TOPICS IN HARMONIC ANALYSIS WITH APPLICATIONS TO …miller/radarla.pdf · TOPICS IN HARMONIC...

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TOPICS IN HARMONIC ANALYSIS WITH APPLICATIONS TO RADAR AND SONAR Willard Miller October 23, 2002

Transcript of TOPICS IN HARMONIC ANALYSIS WITH APPLICATIONS TO …miller/radarla.pdf · TOPICS IN HARMONIC...

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TOPICS IN HARMONIC ANALYSISWITH

APPLICATIONS TO RADAR AND SONAR

Willard Miller

October 23, 2002

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These notes are an introduction to basic concepts and tools in group represen-tation theory, both commutative and noncommutative, that are fundamental for theanalysis of radar and sonar imaging. Several symmetry groups of physical interestare treated (circle, line, rotation, ������� , Heisenberg, etc.) together with their asso-ciated transforms and representation theories (DFT, Fourier transform, expansionsin spherical harmonics, wavelets, etc.). Through the unifying concepts of grouprepresentation theory, familiar tools for commutative groups, such as the Fouriertransform on the line, extend to transforms for the noncommutative groups whicharise in radar-sonar. The insight and results obtained will are related directly to

objects of interest in radar-sonar, such as the ambiguity function. The material ispresented with many examples and should be easily comprehensible by engineersand physicists, as well as mathematicians.

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Contents

0.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 THE DOPPLER EFFECT 51.1 Wideband and narrow-band echos . . . . . . . . . . . . . . . . . 51.2 Ambiguity functions . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 A GROUP THEORY PRIMER 122.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . 122.2 Group representations . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Schur’s lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Orthogonality relations for finite group representations . . . . . . 262.5 Representations of Abelian groups . . . . . . . . . . . . . . . . . 312.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 REPRESENTATION THEORY FOR INFINITE GROUPS 343.1 Linear Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Invariant measures on Lie groups . . . . . . . . . . . . . . . . . . 363.3 Orthogonality relations for compact Lie groups . . . . . . . . . . 413.4 The Peter-Weyl theorem . . . . . . . . . . . . . . . . . . . . . . 443.5 The rotation group and spherical harmonics . . . . . . . . . . . . 473.6 Fourier transforms and their relation to Fourier series . . . . . . . 513.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 REPRESENTATIONS OF THE HEISENBERG GROUP 554.1 Induced representations of ��� . . . . . . . . . . . . . . . . . . . 554.2 The Schrodinger representation . . . . . . . . . . . . . . . . . . . 574.3 Square integrable representations . . . . . . . . . . . . . . . . . . 584.4 Orthogonality of radar cross-ambiguity functions . . . . . . . . . 61

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4.5 The Heisenberg commutation relations . . . . . . . . . . . . . . . 634.6 The Bargmann-Segal Hilbert space . . . . . . . . . . . . . . . . . 674.7 The lattice representation of ��� . . . . . . . . . . . . . . . . . . 724.8 Functions of positive type . . . . . . . . . . . . . . . . . . . . . . 744.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 REPRESENTATIONS OF THE AFFINE GROUP 815.1 Induced irreducible representations of ��� . . . . . . . . . . . . . 815.2 The wideband cross-ambiguity functions . . . . . . . . . . . . . . 845.3 Decomposition of the regular representation . . . . . . . . . . . . 865.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 WEYL-HEISENBERG FRAMES 916.1 Windowed Fourier transforms . . . . . . . . . . . . . . . . . . . 916.2 The Weil-Brezin-Zak transform . . . . . . . . . . . . . . . . . . . 936.3 Windowed transforms and ambiguity functions . . . . . . . . . . 966.4 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.5 Frames of � � � type . . . . . . . . . . . . . . . . . . . . . . . 996.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7 AFFINE FRAMES AND WAVELETS 1037.1 Continuous wavelets . . . . . . . . . . . . . . . . . . . . . . . . 1037.2 Affine frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8 THE SCHRODINGER GROUP 1098.1 Automorphisms of ��� . . . . . . . . . . . . . . . . . . . . . . . 1098.2 The metaplectic representation . . . . . . . . . . . . . . . . . . . 1128.3 Theta functions and the lattice Hilbert space . . . . . . . . . . . . 1178.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

0.1 INTRODUCTION

These notes are intended as an introduction to those basic concepts and tools ingroup representation theory, both commutative and noncommutative, that are fun-damental for the analysis of radar and sonar imaging. Several symmetry groups ofphysical interest will be studied (circle, line, rotation, Heisenberg, affine, etc.) to-gether with their associated transforms and representation theories (DFT, Fourier

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transforms, expansions in spherical harmonics, Weyl-Heisenberg frames, wavelets,etc.) Through the unifying concepts of group representation theory, familiar toolsfor commutative groups, such as the Fourier transform on the line, extend to trans-forms for the noncommutative groups which arise in radar and sonar.

The insights and results obtained will be related directly to objects of interestin radar-sonar, in particular, the ambiguity and cross-ambiguity functions. (Wewill not, however, take up the study of tomography, even though this field hasgroup-theoretic roots.) The material is presented with many examples and shouldbe easily comprehensible by engineers and physicists, as well as mathematicians.

The main emphasis in these notes is on the matrix elements of irreduciblerepresentations of the Heisenberg and affine groups, i.e., the narrow and wideband ambiguity and cross-ambiguity functions of radar and sonar. In Chapter2 we introduce the ambiguity functions in connection with the Doppler effect.Chapters 3 and 4 constitute a minicourse in the representation theory of groups.(Much of the material in these chapters is adapted from the author’s textbook[[79]] .) Chapters 5 and 6 specialize these ideas to the Heisenberg and affinegroups. Chapters 7 and 8 are devoted to frames associated with the Heisenberggroup. (Weyl-Heisenberg) and with the affine group (wavelets). We conclude witha chapter touching on the Schrodinger group and the metaplectic formula.

It is assumed that the reader is proficient in linear algebra and advanced cal-culus, and some concepts in functional analysis (including the basic properties ofcountable Hilbert spaces) are used frequently. (References such as [[1]], [[63]],[[71]], [[86]] and [[95]] contain all the necessary background information.) Thetheory presented here is largely algebraic and (sometimes) formal so as not to ob-scure the clarity of the ideas and to keep the notes short. However, the neededrigor can be supplied. (The knowledgeable reader can invoke Fubini’s theoremwhen we interchange the order of integration, the Lebesgue dominated conver-gence theorem when we pass to a limit under the integral sign, etc.)

Finally, the author (who is not an expert on radar or sonar) wishes to thankthe experts whose writings form the core of these notes, e.g., [[6]], [[8]], [[32]],[[43]], [[55]], [[84]], [[98]], [[113]].

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Chapter 1

THE DOPPLER EFFECT

1.1 Wideband and narrow-band echos

We begin by reviewing the Doppler effect as it relates to radar and sonar. Considera stationary transmitter/detector and a moving (point) target located at a distance�������

from the transmitter at time�. We assume that the distance from the trans-

mitter to the target changes linearly with time,��������� ��� �� Now suppose that the transmitter emits an electromagnetic pulse � ����� . The pulseis transmitted at a constant speed � in the ambient medium (e.g., air or water)impinges the target, and is reflected back to the transmitter/receiver where it isdetected as the echo ����� � ����� . (We assume ����� � � .) Let ����� � ����� be the time delayexperienced by the pulse which is received as an echo at time

�. (Thus this pulse

is emitted at time� ��� and travels a distance ��� before it is received as an echo

at time�.) It follows that the pulse impinges the target at time

� ���! #" and wehave the identity ��� � " ����� �$�! #" ��� " � " ��� �%�! #" � � Thus ����� � �����&� " �� ��� � � " � �%� (1.1)

The echo �'��� � ����� is proportional to the signal at time� �%����� � ����� :

�'��� � �����&� � ( � � �� ��� �*) �,+ � � �� ���.- � " � �%�0/ (1.2)

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Here the ��� factor in the amplitude of the echo is based on the boundary condi-tions for the normal component of the electric field at the surface of a (conducting)target, [[57]]. The factor

� ��� ���� � is needed if we require that the energy of the pulseis conserved: �� � � ����� � � � � � ��� � � ����� � ��� �� Changing notation, we write the echo as����� � ����� � ��� � � � ��� � � ��� � (1.3)

where � � � � � � � � � ��� � and �� � " � � � � � . Note that the transformation� ��� � ��� �� � � � is one-to-one. The result (1.2) is the exact (wideband) solution

for the given assumptions.It is very common to approximate the wideband solution under the assumption

that � � ��� � and�

is small (of the order of � ) over the period of observation. (See

[[28]] and [[104]] for careful analyses of the relationship between the widebandsolution and the narrow-band approximation.) One writes the signal in the form� ����� � � ����� � ���� "!$#�%where �

�����is assumed to be a slowly varying complex function of

�(the envelope

of the waveform) with respect to the exponential factor.Then we have ����� � �����&� � � � � � ��� � � ��� � � ���& (' �&) % � �+*", ! #

or, since �.-/� ��"�0 � � � - � � " � � � � �10 � where 0 � � � , we have����� � ����� - � � ��� ��" � � � ���� 2! # ) %3'54 � ��6 , � �7' �98 � , ':4 � 6 ,;*- � � �=< �& ;6&! # ) �98 � * � ��� ��" � � � �=< �� ;6>! # ) % � � �98 � * (1.4)

This is called the narrow-band approximation of the Doppler effect. (Generallyspeaking, the narrow-band approximation is usually adequate for radar applica-tions but is less appropriate for sonar.) NOTE: We will see that the constant factorof modulus one, � � �=< �� 26&! # ) �98 � * , can be ignored, because it factors out of all in-tegrals and, ultimately, we will be interested only in the absolute values of thecorrelation integrals.

To see the significance of this approximation, consider the Fourier transformof the signal and the echo: ? �A@ � � ��� � ����� � � ���� "!�% � � � (1.5)

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where, [[39]], � ����� � � � ? �3@ � � ���& 2! % � @ (1.6)

Then? �A@ � � � �3@ � � � ? �3@ � "�0 @�� � � �=< �& 2! �98 � (1.7)

in the narrow-band approximation, whereas the exact result is? �3@ � � � �A@ � � � �� �?�� @

��� � ���& 2! � (1.8)

Clearly (1.7) is a good approximation to (1.8) if the support of

? �A@ �lies in a

narrow band around@��

. In the narrow-band approximation the signal is delayedby the time interval " � and the frequency changes by " @� � � .1.2 Ambiguity functions

To determine the position and velocity of a target from a narrow-band echo (1.4)one computes the inner product (cross-correlation) of the echo with a test signal

�'�� � �� �����&� � � �=< �� 2! # ) � � 8 �� * � � � � " ��� � � �=< �& 2! # � �� % � �� ��� 8 � �� �'��� � � �'�� � ���� � � � �'��� � ����� �'�� � �� ����� � �� �������! #� � � #"$� � � "��� � � � ��� �%" � � �� � � " �� � � ���%�& #� ) ' � � � , % * � �

Considering � � ����� � � ���' � ���� � as a function of � � � � , one tries to maximize this function

for a signal � �����)(+* � � � � , the square integrable functions on the real line. Assumethat � is normalized to have unit energy: � � � � � � � � � � � � � � � . Then by the Schwarzinequality we have

� � �'��� � � �'�� � ���� �-, � � �'��� � � �/. � � �'�� � �� � � � � Furthermore the maximum is assumed for � ��� �10 � 32 �'� � � where 4 is a real con-stant. Hence, the maximum is assumed if and only if

� � � � � � � .We can simplify the notation by remarking that5 ��6 �87 � � � � �'��� � � �'� � � � � � � � 9;: �<6=� � 6 �87 � � 7 � � �

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where 9 : ��6 �87 � � � � �*) � � 6" / �*) � �

6" / � < �& ;% � � �� � �

? �3@ � 7 � ? �A@ ��7 � � � ���� "! � � @ �6 � � " � � 6 � " �� � 7 � � @�� �� � 7 � @�� � �� Here, 9 : ��6 �87 � is known as the radar (self-) ambiguity function.

To determine the velocity from the ambiguity function one typically choosesa single-frequency signal of the form � �����&����� ����� � ���� 2!�� % where � @ � 4� and

��� �����&� � � ��� �� , � ,�� ����������� ��� � It can be shown in this case that as a function of � � the ambiguity function hasa sharp peak about � � � � , whereas it is relatively insensitive to changes in

�.

To estimate the range one typically chooses � ����� � �! ����� � ���� 2! #�% where �" �����is a very peaked Gaussian wave function, centered at

� � �and with standard

deviation � � � . This gives an ambiguity function with a sharp peak about/� � , but which is relatively insensitive to changes in � � .

Now we consider the wideband case and generalize to allow a distribution ofmoving targets with density function # � ��� � � , where the support of this functionis contained in the set $ � ��� � �&% � � �(' . Then from (1.3) the echo from the signal� ����� is � �����&� ��� �� � � � � ��� � � ��� � # � ��� � � � � � � (1.9)

Correlating the echo with a “test” echo � ������� generated from the signal � ����� andthe “test” density function # ��� ��� � � we obtain� � � � � � � � � � � � � � � 9 � ��� � ��)���*)� � # � ��� � � # � � )� �*)� � � � � � � )� � )� (1.10)

where 9 � ��� � ��)� �+)� �&� � � � � )� � � � � � � ��� � � � )� � � �,)� � � � � (1.11)

is the radar (self-) ambiguity function in the wideband case. (A very similarconstruction of a moving target distribution can be carried out in the narrow-bandcase.) Note that if # ��� )���*)� �&�.- � )� � � 4 �*)��� � 4 � and # � ��� � � �.- � � � � � � � �1� ���then � � � � � � � � � � � � � � 4 � � 4 � � 9 � � � � � � � � 4 � � 4 �� (1.12)

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As with the narrow-band ambiguity function, if we normalize the signal � to haveunit energy, then by the Schwarz inequality

� 9 � ��� � � )���*)� � � , � � � � � . � � � � � � � (1.13)

and the maximum is assumed for � � )� � � � )� . Note also that

9 � ��� � � )���*)� � � 9 � � � � )� � )� � � )� � � � � � Thus to study the ambiguity function one can restrict to the case9 � ��� � � � 9 � ��� � � � � � � � � � ��� � � � � � � ��� � � ����� � �� (1.14)

Suppose $ � � ' is a basis for* � � � � , the standard space of square integrable

functions on the real line. Then we can consider the cross-correlation of ��� withthe echo � � from � � :9 � �

� ��� � � � � � � � ���)� � � � � � � � � ��� � � ��� � ��� ����� � �� (1.15)

We call 9 � � the cross-ambiguity function of � � and ��� . Similarly the(narrow-band) cross-ambiguity function is

�� ��<6 �87 � � � � � � ) � � 6

" / ��� ) � � 6" / � < �& ;% � � ��

The ambiguity and cross-ambiguity functions are of fundamental importancein the theory of radar/sonar. The use of the ambiguity function to estimate therange and velocity of point targets and, more generally, of the cross-ambiguityfunction to estimate target distribution functions in both the wideband and narrowband cases is basic to the theory.

In these lectures we shall summarize and elucidate this theory by exploitingthe intimate relationship between the cross-ambiguity functions defined above andthe theory of group representations. In particular the wide-band cross-ambiguityfunctions can be interpreted as matrix elements of unitary irreducible represen-tations of the two-dimensional affine group; the narrow-band cross-ambiguityfunctions are matrix elements of unitary irreducible representations of the three-dimensional Heisenberg group. The basic properties of these functions thus emergeas consequences of analysis on affine and Heisenberg groups.

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1.3 Exercises

1. Consider the Gaussian pulse

� �����&� � "� � ���� � � % 8 � � ���& 2! # % �normalized to have unit energy. Verify that the ambiguity function is givenby 9 : ��6 �87 � � � � � ��� � � < � � � �� � � ���& 2! # � Describe the level curves � 9 : ��6 �87 � � �

in the6 � 7 plane. Discuss the

effect of varying the pulse length on the problem of estimating the rangeand velocity of the target.

2. Show that the area enclosed by a level curve � 9 : ��6 �87 � � ��in Exercise 2.1

is independent of .

3. Consider the normalized frequency modulated pulse

� �����&� � "� � ���� � � % 8 � � < �& (' ! # % � � % , The ambiguity function is given by

9;: �<6 �87 � � � � � � '54 � 4���� � � � , � � ��� � � � � � � < � � � � � � ���& 2!$# � Describe the level curves � 9 : �<6 ��7 � � �

in the6 � 7 plane. Discuss

the effect of varying the pulse length and the “compression ratio” � �� � � ��� � ��� � < on the problem of estimating the range and velocity of thetarget.

4. Consider the rectangular pulse with unit energy

� ������� �� " ��� ����� � ���� 2! � %where ��� ������� � � ��� � , � ,�� ��� ��� � � � � �

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Show that the ambiguity function is

9;: �<6 �87 � � � � ����� � � � � � �� � �'��� � 7 � � � � 7 � �,� 6 � , " � � �,� 6 � �" (This isn’t easy.) Describe the level curves � 9 : �<6 ��7 � � �

in the6 � 7

plane. Show that for � � � � � with � very close to zero, the level curves

can be approximated by� � �� � � �

� � � 7 � � � � � .

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Chapter 2

A GROUP THEORY PRIMER

2.1 Definitions and examples

A group is an abstract mathematical entity which expresses the intuitive conceptof symmetry.

Definition 1 A group � is a set of objects $�� ��� � � . . . ' (not necessarily count-able) together with a binary operation which associates to any ordered pair ofelements � ��� in � a third element � � . the binary operation (called group multi-plication) is subject to the following requirements:

1. There exists an element � in � called the identity element such that � � ���� � � for all �( � .

2. For every �( � there exists in � an inverse element �

� 4 such that ���� 4 �

�� 4 � � � .

3. Associative law. The identity�� �� !�

�� � � is satisfied for all � ��� � ( � .

Any set together with a binary operation which satisfies conditions (1) - (3)is called a group. If � �

� � � we say that the elements � and � commute. If allelements of � commute then � is a commutative or abelian group. If � hasa finite number �

� � � of elements it has finite order; otherwise � has infiniteorder.

A subgroup � of � is a subset which is itself a group under the group multi-plication defined in � . The subgroups � and $ � ' are called improper subgroupsof � ; all other subgroups are proper. It can be shown that the identity element �is unique. Also, every element � of � has a unique inverse �

� 4 .12

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Examples 1 1. The real numbers�

with addition as the group product. Theproduct of � 4 � � � ( � is their sum � 4 � � � . The identity is

�and the inverse of

� is � � . Here,�

is an infinite abelian group. Among the proper subgroupsof�

are the integers and the even integers.

2. The nonzero real numbers in�

with multiplication of real numbers as the(commutative) group product. The identity is � and the inverse of � is � � .The positive real numbers form a proper subgroup.

3. The set of matrices � � � � � � �� � � % � ( ���with matrix multiplication as the group product. The identity element is theidentity matrix. Here,� � � 4� � �

� � � �� � � �� � � 4 � � �� � �

so the one-to-one mapping

��� � � � �&� � � �� � � ( � �relating the group element � in

�to

in� �

takes products to products. (Aone-to-one mapping from a group � onto a group � � which takes productsto products is called a group isomorphism. We can identify the two groupsin the sense that they have the same multiplication table.) Thus

�and

� �are isomorphic groups.

More generally we define a homomorphism � % � � � � as a mappingfrom the group � into a group � � which transforms products into products.Thus to every �

( � there is associated � � � � ( � � such that � � � 4 � � �!�� � � 4 � � � � � � for all � 4 � � � ( � . (It follows that � � � � � � � where � � is theidentity element in � � , and � � � � 4 � � � � � � � 4 .) If � is one-to-one and onto(i.e. if � � � � � � � ) then � is an isomorphism of � and � � .

4. The symmetric group

?� . Let � be a positive integer. A permutation �

of � objects (say the set � � $=� � " � . . . � � ' ) is a one-to-one mapping of �onto itself. We can write such a permutation as

� �� � " . . . �� 4 � � . . . � � � (2.1)

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where � is mapped into � 4 , 2 into � � , . . . , � into � � . The numbers � 4 � . . . � � �are a reordering of � � " � . . . � � , and no two of the � � are the same. Theorder in which the columns of (2.1) are written is unimportant. The inversepermutation � � 4 is given by

� � 4 �� � 4 � � . . . � �� " . . . � �

and the product of two permutations � and� � �� 4 � � . . . � �� " . . . � �

is the permutation � �&��� 4 � � . . . � �� 4 � � . . . � � �

(Here we read the product from right to left:�

maps � to � and � maps � to� , so � � maps � to � .) The identity permutation is

� �� � " . . . �� " . . . � �

It is straightforward to show that the permutations of � objects form a group?� of finite order �

� ?�� �

��� . For � ��" �?� is not commutative.

5. The real general linear group � * � � � � � . The group elements 9 are non-singular ��� � matrices with real coefficients:

� * � � � � �&� $/9 � � 9 � � � � ,�� � , �% 9 � ( ��� �� � ��� 9��� � '

Group multiplication is ordinary matrix multiplication. The identity elementis the identity matrix

� � � - � � where- � is the Kronecker delta. The inverse

of an element 9 is its matrix inverse. This group is infinite and for ��� "it is non-abelian. Among the subgroups of � * � � � � � are the real speciallinear group ? * �

� � � �&� $/9 ( � * � � � � � % ��� 9 � � 'and the orthogonal group

� ��� � $/9 ( � * � � � � � % 9;9 % � � '

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where 9 % � � 9 � � is the transpose of the � � � matrix 9 � � 9 � � .Similarly the complex general linear group

� * � � � � �&� $/9 � � 9 � � � � ,�� � , �% 9 � ( � �

�� � � � 9��� � ' �where

�is the field of complex numbers, is a group under matrix multipli-

cation. Among its subgroups are the complex special linear group? * �� � � �&� $/9 ( � * � � � � � % � � 9 � � ' �

and the unitary group� �

��&� $/9 ( � * � � � � � % 9 9 % � � '

where 9 � $ 9 � ' is the complex conjugate of 9 � $/9 � ' . For �� � ,� � � �&� $�� % ��� � � � ' is the circle group.

It is not difficult to show that every finite group is isomorphic to a subgroupof

?� . Furthermore, every finite group is isomorphic to a (finite) subgroup

of � * � � � � � .6. The group � � . This is the finite abelian group whose elements are the in-

tegers� � � � " � . . . � � � � for fixed � � � , and group multiplication is ad-

dition mod � . Thus�

is the identity element and � � is the inverse of � � � � ' � � � � . This group is isomorphic to the multiplicative group of� � � matrices

��� $ ��� � " � � � � %��� � � � � . . . � � � � ' �

where � � ��� � " � � � �is the isomorphism.

7. The affine or ��� � � group. The affine group ��� is the subgroup of � * � � � � �consisting of matrices� ��� � �&� �

� �� � � � ��� � ( � � � � � Clearly the group product is� � 4 � � 4 � . � � � � � � � �

�� 4 � � � 4 � � � � 4� � �

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� � � 4 � � � � 4 � � � � 4 � �the identity element is

� � � � � , and the inverse of� ��� � � is

� � ��� � � � � .8. The (three-dimensional real) Heisenberg group

� �� ��� �� � � 4 � � � � � � �&����� � � 4 � �� � � �� � �

�� % � ( � � ���� This is a subgroup of � * ��� � � � with group product� � 4 � � � � � � � . � � 4 � � � � � � �&� � � 4 � � 4 � � � � � � � � � � � � � � 4 � � �� The identity element is the identity matrix

� � � � � � � and� � 4 � � � � � � � � 4 �� � � 4 � � � � � � 4 � � � � � � . Note that � � is non abelian. The center of ��� ,

i.e., the subgroup�� of all elements which commute with every element of

� � , �� � $ � ( � � % � � � � � �� � ( � � '

consists of the elements� � � � � � � � � � � ( �

. A finite analog of � � is

� �� ��� �� ��� � � 4 � �� � � �� � �

� � % � ( � � � ���� �a finite group under matrix multiplication where addition and multiplicationof the number � is carried out mod � .

9. Let � be a finite dimensional vector space, real or complex, and denoteby � * � � � the set of all invertible linear transformations of � of � onto� . Then � * � � � is a group with the product of � 4 � � � ( � * � � � given by� 4 � � where

� � 4 � � ��� � � 4 � � � � � for each� ( � , i.e. , � 4 � � is the usual

product of linear transformations. The identity element is the operator �such that � �%���

for all� ( � . The inverse of � ( � * � � � is the inverse

linear operator � � 4 where � � ��� for� � � ( � if and only if � � 4 � ���

.

Suppose ������ �� and let

� 4 � . . . � � � be a basis for � . The matrixof the operator � with respect to the basis is � � � � where � � � �� � ! 4 � � � � , , � . It is easy to see that this correspondence defines anisomorphism between the group � * � � � and � * � � � .Many of the most important applications of groups to the sciences are ex-pressed in terms of the group representation, to which we now turn.

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2.2 Group representations

Definition 2 A representation (rep) of a group � with representation space �is a homomorphism � %

�� � � � � of � into � * � � � . The dimension of the

representation is the dimension of � .

It follows from this definition that

� � � 4 � � � � � � � � � � 4 � � � � � � � � � 4 � � � � � 4 � �� � � �&��� � ��� ����� ��� (� � (2.2)

(Initially we shall consider only finite-dimensional reps of groups on complex repspaces. Later we will lift these finiteness restrictions.)

Definition 3 An n-dimensional matrix rep of � is a homomorphism %��

� � � of � into � * � � � � � .The � � � matrices � � � � � ( � , satisfy multiplication properties analogous

to (2.2). Any group rep � of � with rep space � defines many matrix reps, sinceif $ � 4 � . . . � � � ' is a basis of � , the matrices � � � � � � � �� � � defined by

� � � ����,� � � 4 � � � � � � � � , , � (2.3)

form an � -dimensional matrix rep of � . Every choice of basis for � yields anew matrix rep of � defined by � . However, any two such matrix reps � � areequivalent in the sense that there exists a matrix

? ( � * � � � � � such that

� � � � � ? � � �

? � 4 (2.4)

for all �( � . Indeed, if � � correspond to the bases $ � ' � $ � � ' respectively,

then for

?we can take the matrix

� ? � � defined by� � � � 4

?� � �� � � � � � . . . � � (2.5)

Definition 4 Two complex � -dimensional matrix reps and � are equivalent� -. � � if there exists an

? ( � * � � � � � such that (2.4) holds.

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Thus equivalent matrix reps can be viewed as arising from the same operatorrep. Conversely, given an � -dimensional matrix rep � � � we can define many � -dimensional operator reps of � . If � is an � -dimensional vector space with basis$ � ' we can define the group rep � by (2.3), i.e., we define the operator � � � � bythe right-hand side of (2.3). Every choice of a vector space � and a basis $ � 'for � yields a new operator rep defined by . However, if � � � �

are two such� -dimensional vector spaces with bases $ � ' � $ � � ' respectively, then the reps �and � � are related by � � � � � ��� � � � ��� � 4 � (2.6)

where�

is an invertible operator from � onto � � defined by

� � ��� � � ,�� , �

Definition 5 Two � -dimensional group reps � � � �of � on the spaces � � � � are

equivalent� ���� � � � if there exists an invertible linear transformation

�of �

onto � � such that (2.6) holds.

Clearly, there is a one-to-one correspondence between classes of equivalentoperator reps and classes of equivalent matrix reps. In order to determine allpossible reps of a group � it is enough to find one rep � in each equivalenceclass. It is a matter of choice whether we study operator reps or matrix reps.

Examples 2 1. The matrix groups � * � � � � � � ? * � � � � � are � -dimensional ma-trix reps of themselves.

2. Let � be a finite group of order � . We formally define an � -dimensionalvector space

���consisting of all elements of the form

��� �

� � � � . � � � � � �)( � Two vectors

� � � � � . � and� � � � � . � are equal if and only if �

��� � � � � �

for all �( � . The sum of two vectors and the scalar multiple of a vector

are defined by� � � � � . � � � � � � � . � � � � � � � � � � � � � � . � �4 � � � � � . � � � 4 � � � � . � � 4 ( � (2.7)

The zero vector of��

is � � � . � . The vectors � . � � , � � ( � , form anatural basis for

��. (From now on, we make the identification � .�� � �

(18

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� �.) We define the product of two elements �

� � � � � � . � � � � � � � � � . �in a natural manner:

� � � � � � � � � .�� � � � � � � � . � �&� � � � � � � � � � � � � � � . � �� � � � � � � � . � (2.8)

where� � �� � �

� � �� � � � � � � � 4 �� (2.9)

(Here � � � � � is called the convolution of the functions ���� � � � � � .) It is easy

to verify the following relations:� � � � � � � � � � � � � � � � � � �&� � � � � � � ��� � � � ( ��� �� � � � � � � � �� � � 4 � � � � � � 4 � � � � � � 4 � � �� � � � � � ��� 4 ( � � (2.10)

where � is the identity element of � . Thus� �

is an algebra, called thegroup ring of � .

The mapping�

of � into � * � ��0� given by� �

�� � � � ��� � ( ���

(2.11)

defines an � -dimensional rep of � , the (left) regular rep. Indeed,� �

� 4 � � � � � � 4 � � � ��� � � 4 � � � � ��� � � 4 ��� � � � � �� � � � � � � � � �

and the� �

��

are linear operators. Similarly, the (right) regular rep of �is defined by � �

�� � � � �

� 4 � � ( ��� � �( � (2.12)

Let � be a rep of the finite group � on a finite-dimensional inner productspace � . The rep � is unitary if for all �

( �� � � � ��� � � � � ��� � � � � � � � � � � � ( � � (2.13)

i.e., if the operators � � � � are unitary. Recall that an orthonormal (ON) basis forthe � -dimensional space � is a basis $ � 4 � . . . � � � ' such that

� � � � � � �.- � , where� . � .3� is the inner product on � . The matrices � � � of the operators � � � � withrespect to an

���basis $ � ' are unitary matrices � � � � � � � � 4 � � � � � � � � 4 � �

Hence, they form a unitary matrix rep of � . The following theorem shows thatfor finite groups at least, we can always restrict ourselves to unitary reps.

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Theorem 1 Let � be a rep of � on the inner product space � . Then � is equiv-alent to a unitary rep on � .

PROOF: First we define a new inner product� . � . � on � with respect to which �

is unitary. For � � � ( � let�� � � �&� �

�� � � ��� � � � � � � � � � � � � � � (2.14)

(Thus�� � � � is an average of the numbers

� � � � � � � � � � � � � taken over the group.)It is easy to check that

� . � . � is an inner product on � . Furthermore,� � � � � � � � � � � � � � 4� ' � , � � � � � � � � � � � � � � � � ��� �� 4� ' � , � � � � � � � � � � � � � � � � ��� �� �� � � � �

where the next to last equality follows from the fact that if � runs through theelements of � exactly once, then so does � � . Clearly, � is unitary with respectto the inner product

� . � . � . Now let $�� ' be an���

basis of � with respect to� . � . � and let $ � ' be an���

basis with respect to� . � .3� . Define the nonsingular

linear operator� % � � � by

�� ��� � � , � , � . Then for

� � � 4 � and� � � 0 � � � we find � � � � � � � � � � � 4 0 � � � � � � � � �� � 4 0 � � � ��� �so

� � ��� � � � � � 4 � � � � 4 � � and� � � � � ��� � 4 � � � � � � ��� � 4 � � � � � � � � � � 4 � � � � � ��� � 4 � �� � � � 4 � � � � 4 � �&� � � ��� � Thus, the rep � ��� � � ��� � � � � � � 4 is unitary on � . Q.E.D.

It follows that we can always assume that a rep � on � is unitary.Now we study the decomposition of a finite-dimensional rep of a finite group

� into irreducible components.

Definition 6 A subspace � of � is invariant under � if � � � ��� ( � for every�( � ,

� ( � .

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If � is invariant we can define a rep � � � � � � of � on � by

� � � � � � � � � � � � � � ( � This rep is called the restriction of � to � . If � is unitary so is � �

.

Definition 7 The rep � is reducible if there is a proper subspace � of � whichis invariant under � . Otherwise, � is irreducible (irred).

A rep is irred if the only invariant subspaces of � are $ ' , the zero vector, and� itself.Every reducible rep � can be decomposed into irred reps in an almost unique

manner. In proving this, we can assume that � is unitary.If � is a proper subspace of the inner product space � and

�� � $ � ( � % � � � � � � � � ����� � ( � '

(2.15)

is the subspace of all vectors perpendicular to � , it is easy to show that � �� � �

�, ( � is the direct sum of � and �

�). That is, every

� ( � can bewritten uniquely in the form� ��� � � � � � ( � � � � ( � � Theorem 2 If � is a reducible unitary rep of � on � and � is a proper invariantsubspace of � , then �

�is also a proper invariant subspace of � . In this case we

write � � � � ��� � � and say that � is the direct sum of � � and � � � , where � � � � � �are the (unitary) restrictions of � to � � � �

, respectively.

PROOF: We must show � � � � � ( � �for every �

( � � � ( � �. Now for every� ( � , � � � � � � � � � � �

� � � � � � 4 � � � � �since � � � � 4 ��� ( � . The first equality follows from (3.13) and unitarity. Thus� � � � � ( � �

. Q.E.D.Suppose � is reducible and � 4 is a proper invariant subspace of � of smallest

dimension. Then, necessarily, the restriction � 4 of � to � 4 is irred and we havethe direct sum decomposition � � � 4 � � �4 , where � �4 is invariant under � . If� �4 is not irred we can find a proper irred subspace � � of smallest dimension such

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that � �4 � � � � � �� , by repeating the above argument. We continue in this fashionuntil eventually we obtain the direct sum decomposition� � � 4 � � � � . . . � � � ��� � � � 4 ��� � � . . . ��� �where the � are mutually orthogonal proper invariant subspaces of � which trans-form irreducibly under the restrictions � of � to � . (The decomposition processcomes to an end after a finite number of steps because � is finite-dimensional.)Some of the � may be equivalent. If � 4 of the reps � are equivalent to � 4 � � � to� � � . . . � � to � and � 4 � . . . � � are pairwise nonequivalent, we write

� � � 4 � � � � � (2.16)

Theorem 3 Every finite-dimensional unitary rep of a finite group can be decom-posed into a direct sum of irred unitary reps

The above decomposition is not unique since the irred subspaces � 4 � . . . � � �are not uniquely determined. However, it can be shown that the integers � � in(2.16) are uniquely determined, [[20]], [[43]], [[79]].

2.3 Schur’s lemmas

The following two theorems (Schur’s lemmas) are crucial for the analysis of irredreps.

Theorem 4 Let � � � � be irred reps of the group � on the finite-dimensional vec-tor spaces � � � � respectively and let

�be a nonzero linear transformation map-

ping � into � � , such that � � � � � � � � � � � � (2.17)

for all �( � . Then

�is a nonsingular linear transformation of � onto � �

, so �and � � are equivalent.

PROOF: Let���

be the null space and� �

the range of�

:��� � $ � ( � % � � � ' � � � � $ � � ( � � % � � � � � � � � � � � � � ( � '

The subspace� � of � is invariant under � since

� � � � � � � � ��� � � � � � forall �

( � � � ( � � . Since � is irred,���

is either � or $ ' . The first possibility

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implies�

is the zero operator, which contradicts the hypothesis. Therefore,� � �

$ ' . The subspace� �

of � � is invariant under � � because � � � � � � � � � � � � � � (� �for all

� ( � . But � � is irred so� �

is either � � or $ ' . If� � � $ ' then

is the zero operator, which is impossible. Therefore� � � � � which implies that� and � � are equivalent. Q.E.D.

Corollary 1 Let � � � � be nonequivalent finite-dimensional irred reps of � . If�

is a linear transformation from � to � � which satisfies (2.17) for all �( � , then�

is the zero operator.

While Theorems 1 – 4 and Corollary 1 apply also for real vector spaces, thefollowing results are true only for complex reps.

Theorem 5 Let � be a rep of the group � on the finite-dimensional complex vec-tor space � ,

� � � � � � � . Then � is irred if and only if the only transformations� % � � � such that � � � � � � � � � � � (2.18)

for all �( � are

� � � � where� ( �

and � is the identity operator on � .

PROOF: It is well known that a linear operator on a finite-dimensional complexvector space always has at least one eigenvalue. Let

�be an eigenvalue of an

operator�

which satisfies (2.18) and define the eigenspace���

by

��� � $ � ( � % � � � � � ' Clearly

���is a subspace of � and ���� ��� � �

. Furthermore,���

is invariantunder � because � � � � ��� � � � � � � � � � � � � � �for

� ( ���, �

( � , so � � � � � ( ���. If � is irred then

��� � � and� � � � �

forall� ( � .Conversely, suppose � is reducible. Then there exists a proper invariant

subspace � 4 of � and by Theorem 2, a proper invariant subspace � � such that� � � 4 � � � . Any� ( � can be written uniquely as

� � � 4 � � � with� � ( � � . We define the projection operator � on � by � � � � 4 ( � 4 . Then� � � � ��� � � � � � � � � � � � � � 4 (verify this), and � is clearly not a multiple of � .Q.E.D.

Choosing a basis for � and a basis for � � we can immediately translate Schur’slemmas into statements about irred matrix reps

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Corollary 2 Let and � be � � � and � � � complex irred matrix reps of thegroup � , and let 9 be an � � � matrix such that

� � � � 9 � 9 � � � (2.19)

for all �( � . If and � are nonequivalent then 9 equals the zero matrix. (In

particular, this is true for � �� � .) If 0. � then 9 0 � �� where

� ( �and

��

is the ��� � identity matrix.

Note that the proofs of Schur’s lemmas use only the concept of irreducibil-ity and the fact that the rep spaces are finite-dimensional. The homomorphismproperty of reps and the fact that � is finite are not needed.

Example 1 # < , the symmetry group of the square. Consider the square � 9 ����� � � # �with center at the origin of the � �1� plane and vertices with coordinates9 � ��� � � � ��� � � � � � � � � � � ��� � � # � ��� � ��� �� There are precisely 8 Euclidean transformations (rotations and reflections) thatmap the square to itself. These are the congruences of euclidean geometry. Sincethe square is uniquely determined by the location of its vertices, we can describethe transformations as permutations of the vertices. Thus # < can be consideredas a subgroup of the symmetric group on 4 letters. The symmetries are:� 9 � � #9 � � # � � � � � � � � ���� 9 � � #� � # 9 � � � � ��� � � ��� � ��� � ����� � � � � �� 9 � � #� # 9 � � � � �� � � � � � ��� � ��� �&� ��� � � � � �� 9 � � ## 9 � � � � � "� � � � � � ��� � ��� �&� ��� � � � � �� 9 � � ## � � 9 � � � ��� � � � � � � � ��� � � � � �� 9 � � #� 9 # � � � � � ��� � � � � � � � ��� � � � � �� 9 � � #9 # � � � � � � ��� � � � � � � � ��� �� ��� � � � ������ 9 � � #� � 9 # � � � ��� � � � � � � � ��� �� ��� � � � �����

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It is straightforward to check that these transformations satisfy the group axioms.Recall that the tranformations are applied from right to left. Thus a

� � �clockwise

rotation followed by a reflection in the � axis is

� � � 9 � � ## � � 9 �� 9 � � #� � # 9 � �

� 9 � � #� � 9 # � �the reflection in the diagonal � # .

This realization of # < as a group of transformations in the plane automaticallygives us a 2-dimensional representation of # < . Indeed the

� ���rotation

and the

reflection � correspond to the operators � � � � � � � � in the plane such that

� � � % � 4 � � � �� � � � 4 � � � � � % � 4 � � 4� � � � � � �where � 4 � � � are the unit vectors

� � � � � � � � � � � in the � and � directions, respec-tively. Since

��� generate the full group action, the remaining operators can beconstructed by taking appropriate products of these. Using this ON basis we ob-tain the unitary matrix representation of # < generated by the matrices

� � �� � �� � � � � � � ���

� � �� � ��� �

Now let us consider this matrix representation to be defined over the complexfield, and check to see if it is reducible. If

9 � �� �� � �

then we will have 9 � � � � � � � 9 � � � � ��� � � ( # <if and only if 9 ��#� � ��#� 9 � 9 � � �&� � � � 9 The first of these condition implies �

� � � � � � � , and the second implies ��� � �

. Thus 9 must have the form

9 � �

� � �� � �25

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and is irreducible.Now consider the rotation subgroup

� < � $ � � � � � � ' of # < . If we restrictour representation to this subgroup we get a matrix representation of

� < gener-ated by the matrix � � . This representation will be irreducible provided the onlymatrix 9 that commutes with � � is a multiple of the identity. However, we seefrom our above computation that 9 ��#�&� ��#� 9 for any

9 � �� �

� � � �where � � � are arbitrary. Thus this representation is reducible. In particular theone-dimensional subspace with basis

� � � � � , where � � � ��� , is invariant under� � � .2.4 Orthogonality relations for finite group repre-

sentations

Now let � be a finite group again and select one irred rep � ' � , of � in eachequivalence class of irred reps Then every irred rep is equivalent to some � ' � , andthe reps � ' � � , � � ' � , are nonequivalent if � 4 �� � � . The parameter � indexes theequivalence classes of irred reps. (We will soon show that there are only a finitenumber of these classes.) Introduction of a basis in each rep space � ' � , leads to amatrix rep ' � , . The ' � , form a complete set of irred � � � � � matrix reps of � ,one from each equivalence class. Here � �

� �� � � ' � , . If we wish, we can choosethe ' � , to be unitary.

The following procedure leads to an extremely useful set of relations in reptheory, the orthogonality relations. Given two irred matrix reps ' � , � ' � , of �choose an arbitrary � � � � � matrix � and form the � � � � � matrix9 � � � 4 ��� � ' � , �

�� � ' � , �

�� 4 � (2.20)

where� �

�� � � . (Here, 9 is just the average of the matrices ' � , � � � � ' � , � � � 4 �

over the group � .) We will show that 9 satisfies

' � , � � � 9 � 9 ' � , � � � (2.21)

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for all �( � . Indeed,

' � , � � � 9 � � � 4 � ��� � ' � , � � � ' � , � � � � ' � , � � � 4 �� � � 4 � ��� � ' � , � � � � � ' � , � � � � � � 4 � ' � , � � �� 9 ' � , � � �� We have used the fact that as � runs over each of the elements of � exactly once,so does �

� � � � . This result and Corollary 1 imply that if � ����then 9 is the

zero matrix, whereas if � ���then 9 � � �

��� for some� ( �

. Hence, 9 �� � � � � ��- � �

���� where

-� � is the Kronecker delta, and the coefficient

�depends on� and � .

To derive all possible consequences of this identity it is enough to let � runthrough the � � � � � matrices � ' � � � , � � � ' � � � ,� �

, where

� ' � � � ,� � - � � - � � � , ��� , � �� � , � � , � �

Making these substitutions, we obtain

��� � ' � , � � � � ' � ,

��: � � � 4 �&� � � - � �- : � � ,�� ��� , � �

� � , � � � , � �

(2.22)

Here,�

may depend on � ��� and � , but not on � or � . To evaluate�

, set��� � � � �

� , and sum on � to obtain

� �� � � � ��� � � � � 4 ' � ,

� � � � 4 � ' � , � � � � � � ��� � ' � ,� � � �� � -

� since

� ��� � � and ' � ,

� � � �*� -� . Therefore,

� � -� � � . We can simplify

(2.22) slightly if we assume (as we can) that all of the matrix reps ' � , � � � areunitary. Then ' � ,

��: � � � 4 � � ' � ,: � � � �and (2.22) reduces to

��� � ' � , � � � � ' � ,: � � � � � �

� �

- : - � � - � �

(2.23)

Expressions (2.22), (2.23) are the orthogonality relations for matrix elements ofirred reps of � . We can write these relations in a basis-free manner. Supposethe irred unitary reps � ' � , � � �

act on the rep spaces � ' � , � � ' � , with���

bases

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$ � ' � , ' � $ � ' � ,: ', respectively. Then we have ' � , � � � � � � � ' � , � � � � ' � ,� � � ' � , � with a

similar expression for � ' � , . Expanding arbitrary vectors � ��� ( � ' � , in the basis$ � ' � ,� 'and � � ��� �=( � ' � , in the basis $ � ' � ,: '

and using (2.23) we find

��� �

� � ' � , ��� � ��� � � � � � � ' � , �

�� � � � � �

�� ��� � � � ��� � � - � �

� �

(2.24)

That is, the left hand side of (2.24) vanishes unless � � �, in which case the inner

products on the right hand side correspond to the space � ' � , � � ' � , .To better understand the orthogonality relations it is convenient to consider

the elements � of the group ring��

as complex-valued functions ����

on thegroup � . The relation between this approach and the definition of

� �as given in

example is provided by the correspondence

� � ��� �

� � � � . � � � � � � �� (2.25)

The elements of the�

-tuple� � � � 4 � � . . . � � � ��� ��� , where � ranges over � , can

be regarded as the components of �( � �

in the natural basis provided by theelements of � . Furthermore the one-to-one mapping (2.25) leads to the relations

� � � � � � � � � � � � � � 4 � � 4 � � � �� � � � � � � � � � � � � � � � � � � � � � 4 � � (2.26)

where the expression defining � � � � � is called the convolution product of ����

and � � � � . The ring of functions just constructed is algebraically isomorphic to���with the isomorphism given by (2.26). Under this isomorphism the element

� � � . � ( ���is mapped into the function

� � � � � � � ��� � � �� ��� ��� � � � � � Now consider the right regular rep on

� �. Writing

� � � � � � � � � �

� � � � ��� � � � .�� � � �� 4 �

�� � � � � . � �

we obtain � � � � � ��� � � � � � � � � � � � ( � � (2.27)

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as the action of� � � � on our new model of

��. From Theorem 1, there is an inner

product on the�

-dimensional vector space� �

with respect to which the rightregular rep

�is unitary. Indeed, the following inner product works:� ��� � � � � � 4 ��� � � � � � � � � � � ��� � ( ���

(2.28)

Now note that for fixed � � � � with � , � � , � � the matrix element ' � , � � � �defines a function on � , hence an element of

� �. Furthermore, comparing (2.28)

with (2.23), we see that the functions

� ' � , � � � � � � 4 8 �� ' � , � � � � � � ,�� � , � �� (2.29)

where � ranges over all equivalence classes of irred reps of � , form an� �

setin���

. Since��

is�

-dimensional the� �

set can contain at most�

elements.Thus there are only a finite number, say � , of nonequivalent irred reps of � . Eachirred matrix rep � yields � �� vectors of the form (2.29). The full

���set $ � ' � , � 'spans a subspace of

��of dimension

� � 4 � � �� � . . . � � �� , � (2.30)

The inequality (2.30) is a strong restriction on the possible number and dimensionsof irred reps of � . This result can be further strengthened by showing that the

� �set $ � ' � , � ' is actually a basis for

��. Since the dimension

�of��

is equal to thenumber of basis vectors, we obtain the equality

� � 4 � � �� � . . . � � �� � � (2.31)

To prove this result, let � be the subspace of� �

spanned by the� �

set $ � ' � , � ' .From (2.29) and the homomorphism property of the matrices ' � , � � � there follows

� � � � � � ' � , � � � � � � � ' � , � � � � � �� � 4 '

� , � � � � � ' � , � � �)( � (2.32)

Thus � is invariant under�

. According to Theorem 2, � �is also invariant under�

and��� � � ��� �

. (Here � �is defined with respect to the inner product

(2.28).) If � � �� $ ' then it contains a subspace � transforming under someirred rep � ' � , of � . Thus, there exists an

� �basis � 4 � . . . � � ��� for � such that

� � � � � � � � � �&� � � � � � ���� � 4 '

� ,� � � � � � � � � � � ,���, � �

(2.33)

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Setting � � � in (2.33) we find

� � � � � �� � � � � ' � ,� � � � �

�� � � � � � ' � ,� � � � � 4 8 ��

�so � ( � . Thus � � ��� � �

. This is possible only if � � $ ' . Therefore,� � � $ ' and � � ���.

Theorem 6 The functions

$ � ' � , � � � � ' � � � � � . . . � � � � ,�� � , � ��

form an� �

basis for���

. Every function �( ��

can be written uniquely in theform

� � � �&�

� � � � �� � � ' � , � � � � � � � � � � ��� � ' � , � �� (2.34)

The series (2.34) is the “generalized Fourier expansion” for the functions �(���

and the �� � are the “generalized Fourier coefficients”. Furthermore we have

the “generalized Plancherel formula”� ��� � � �

� � � �� ��� � ' � , � � � � ' � , � � � �� (2.35)

We can also write expansion (2.34) in a basis-free form through the use of (2.29)and the homomorphism property of the matrices ' � , � � � :

� � � � � � � � � � � �� ��� ' � , � � ' � , � � � �� 4� � � � � � � �� � � � � � � � ' � , � � � � ' � , � � � �� 4� � � � � � � �� � � � � � � � ' � ,� � � � 4 � ' � , � � � �� 4� � � � � � �� � � � � � � � ' � ,� � � � � 4 � �� � � � � � �

� ��� ' � ,� � � � � 4 . � �� �� � �

� ��� ��� ' � , � � � 4 . ���Here

� � ' � , � � � is the trace of the matrix ' � , � � � .Relation (2.35) can be written in the basis-free form� ��� � � �

� �� � ' � ,� � � �

where � ' � ,� � � � � � ��� ��� ' � , � � � 4 . ��� .30

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2.5 Representations of Abelian groups

The representation theory of Abelian (not necessarily finite) groups is especiallysimple.

Lemma 1 Let � be an Abelian group and let � be a finite-dimensional irred repof � on a complex vector space � . Then � , hence � , is one-dimensional.

PROOF: Suppose � is irred on � and ������ � � . There must exist a �( �

such that � � � � is not a multiple of the identity operator, for otherwise � would bereducible. Let

�be an eigenvalue of � � � � and let

���be the eigenspace

��� � $ � ( � % � � � ��� � � � ' Clearly,

���is a proper subspace of � . If � ( � and

� ( ���then� � � �'� � � � ��� � � � � � � � � � � � � � � � � � � � � � � �

since � is Abelian, so���

is invariant under the operator � � � � . Therefore, � isreducible. Contradiction! Q.E.D.

Example 2 � � � � � �This is the Abelian group of order

�, example (6) in � 2.1, with addition of

integers � � � as group multiplication. Since � � � � for each irred rep of � � ,it follows from (2.31) that � � has exactly � � � distinct irred reps. Since the

elements of � � can be represented as �� � � � � � � " � . . . � � � � where � �� � � ,

we have � � ' � , � � ��� � � � � ' � , � � �� � � � ' � , � � � � � for each irred rep ' � , . Thus ' � , � � ��� is an�

th root of unity and it uniquely determines � ' � , � � � for all �( � � .

Since there are exactly�

such roots, which we label as� ' � , �����&� ��� � " � � � � � � � � � � � � . . . � � � � �

the possible irred reps are� ' � , ��� � � ��� � " � � �0 � � � � � � � � � . . . � � � � (2.36)

It follows from Theorem 6 that every function �( �����

has the finite Fourierexpansion

� �� �&� � � 4 � ���� � � � ��� � " � � �0 � � (2.37)

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where

�� � � ��� � ��� ' � , � � ��

� � 4 � � � � � � ��� � � " � � � �0 � ��

Furthermore, the orthogonality relations are

� ' � , � ' � , �&�.-� �� ��

� � 4 � � ��� � " � � � � � � � � � � � (2.38)

and the Plancherel formula reads

��

� � 4 � � �� � � �� � � � � 4

� � �� � � � �� � � � � ��� � ( � � � Since finite non-Abelian groups do not occur in the sequel, we refer the reader

to standard textbooks for examples of unitary irred reps of these groups, [[20]],[[79]].

2.6 Exercises

1. The center of a group � is the subgroup

� � $ � ( � % � � � � � � � � ( � ' Compute the center of the Heisenberg group � � and the center of the affinegroup � � .

2. Let � be an � -dimensional complex vector space with basis $ � 4 � . . . � � � ' .The matrix of the operator � ( � * � � � with respect to this basis is �� � � where � � � � � � ! 4 � � , � , , � . Verify that the correspondence� � is an isomorphism of the groups � * � � � and � * � � � .

3. Show that the map �� � �

��

is a rep of the group � on the group ring� �

where� �

�� � � � �

� 4 for �( � , �

( ���.

4. Verify explicitly that the Hermitian form�� � � � , (3.14), defines an inner

product on the vector space � . (Among other things, you must show that�� � � � � �

if and only if �� .)

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5. Let 4 � � � and � � � � be � � � matrix reps of the group � with real ma-trix elements. These reps are real equivalent if there is a real nonsingularmatrix

?such that 4 � � �

? � ? � � � � for all �

( � . Show that 4 and �are complex equivalent if and only if they are real equivalent. (Hint: Write? � 9 � � � , where 9 and � are real, and show that 9 � � � is invertiblefor some real number

�.)

6. Show that the matrix elements of two real irred reps of a group � which arenot real equivalent satisfy an orthogonality relation. Show that every realirred rep is real equivalent to a rep by real orthogonal matrices.

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Chapter 3

REPRESENTATION THEORYFOR INFINITE GROUPS

3.1 Linear Lie groups

We will now indicate how some of the basic results in the rep theory of finitegroups can be extended to infinite groups. A fundamental tool in the rep theory offinite groups is the averaging of a function or operator over the group by taking asum. We will now introduce a new class of groups, the linear Lie groups, in whichone can (explicitly) integrate over the group manifold and, at least for compactlinear Lie groups, can prove close analogs of Theorems 1–5.

Let � be an open connected set containing � � � � � . . . � � � in the space�� of

all (real or complex) � -tuples � � �� 4 � . . . � � � � . (The reader can assume � is an

open sphere with center � .)Definition 8 An � -dimensional local linear Lie group � is a set of � � � non-singular matrices 9 � � � � 9 � � 4 � . . . � � � � , defined for each � ( � , such that

1. 9 � � � � �� (the identity matrix)

2. The matrix elements of 9 � � � are analytic functions of the parameters� 4 � . . . � � � and the map � � 9 � � � is one-to-one.

3. The � matrices� � '�� ,� ��� � � � � . . . � � , are linearly independent for each

� ( � . That is, these matrices span an � -dimensional subspace of the� � -dimensional space of all � � � matrices.

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4. There exists a neighborhood � �of � in

�� � � � � � , with the property that

for every pair of � -tuples � � � in � �there is an � -tuple � in � satisfying9 � � � 9 � � �&� 9 � � � (3.1)

where the operation on the left is matrix multiplication.

From the implicit function theorem one can show that (4) implies that thereexists a nonzero neighborhood � of � such that � � � � � ��� � for all � ��� ( �where � is an analytic vector-valued function of its an arguments and � ��� � � arerelated by (3.1), [[52]], [[96]].

Let � be a local linear group of ��� � matrices. We will now construct a (con-nected, global) linear Lie group

)� containing � . Algebraically,)� is the abstract

subgroup of � * � � � � � generated by the matrices of � . That is,)� consists of all

possible products of finite sequences of elements in � . In addition, the elementsof

)� can be parametrized analytically. If � ( )� we can introduce coordinatesin a neighborhood of � by means of the map � � � 9 � � � where � ranges overa suitably small neighborhood � of � in

�� . In particular, the coordinates of �

will be � � � � � . . . � � � . Proceeding in this way for each � ( )� we can cover)�

with local coordinate systems as “coordinate patches”. The same group element�will have many different sets of coordinates, depending on which coordinate

patch containing�

we happen to consider. Suppose�

lies in the intersection ofcoordinate patches around � 4 and � � , respectively. Then

�will have coordinates

� 4 ��� � respectively, where� � � 4 9 � � 4 � � � � 9 � � � � . Since9 � � 4 �&� � � 44 � � 9 � � � � � 9 � � � �&� � � 4� � 4 9 � � 4 �

it follows that in a suitably small neighborhood of � : (a) the coordinates � � areanalytic functions � � ��� � � 4 � of the coordinates � 4 , (b)

�is one-to-one, and (c)

the Jacobian of the coordinate transformation is nonzero. This makes)� into an

analytic manifold. (In addition to the coordinate neighborhoods described above,we can always add more coordinate neighborhoods to

)� provided they satisfyconditions (a) – (c) on the overlap with any of the original coordinate systems.)We leave it to the reader to show that

)� is connected. That is, any two elements9 ��� in)� can be connected by an analytic curve

� �����lying entirely in

)� .In general an � -dimensional (global) linear Lie group � is an abstract matrix

group which is also an � -dimensional local linear group � . Clearly, � � )� .Indeed,

)� is the connected component of � containing the identity matrix. Thegroup � need not be connected.

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Examples (3), (5) ( � * � � � � � , ? * � � � � � , � � � � , � * � � � � � , ? * � � � � � , � ���),

(7), (8)�� �

�from � 2.1 are all linear Lie groups. To illustrate, we can write 9 (

� * � � � � � for 9 “close” to the identity matrix as 9 � � - � � � � � � , � , � ,and verify that conditions (1) – (4) of the definition of a local linear Lie group aresatisfied for the coordinates � � $�� � ' (%�

� . It follows that � * � � � � � is a realglobal � � -dimensional linear Lie group.

3.2 Invariant measures on Lie groups

Let � be a real � -dimensional global Lie group of � � � matrices. A function� � � � on � is continuous at � ( � if it is a continuous function of the parameters�� 4 � . . . � � � � in a local coordinate system for � at � . (Clearly, if

�is continuous

with respect to one local coordinate system at � it is continuous with respect toall coordinate systems.) If

�is continuous at every � ( � then it is a continuous

function on � . We shall show how to define an infinitesimal volume element � 9in � with respect to which the associated integral over the group is left-invariant,i.e., �

� � � � 9 � � 9 � �� � � 9 � � 9 � � ( � � (3.2)

where�

is a continuous function on � such that either of the integrals converges.In terms of local coordinates � � �

� 4 � . . . � � � � at 9 ,

� 9 � 7 � � � � � 4 . . .+� � � � 7 � � � � � � (3.3)

where the positive continuous function 7 is called a weight function. If � �� � 4 � . . . � � � � is another set of local coordinates at 9 then,

� 9 � )7 � � � � 4 . . .7� � � )7 � � � 7 � � � � � � � ��� ��� � � � � � �where the determinant is the Jacobian of the coordinate transformation. (For aprecise definition of integrals on manifolds see [[101]].)

Two examples of left-invariant integrals are well known. The group� �

(exam-ple (3) in � 2.1) with elements

9 � � �&� � � �� � � � � ( �(3.4)

is isomorphic to the real line. The continuous functions on� �

are just the contin-uous functions

� � � � on the real line. Here, � � is a left-invariant measure. Indeed

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by a simple change of variable we have��� � � � � � � � � � �� � � � � � ��� � ( �where

�is any continuous function on

�such that the integral converges. Since�

is Abelian, � � is also right-invariant.A second example is the circle group

� � � �&� $ � �� ' , the Abelian group of � � �unitary matrices. The continuous functions on

��� � � can be written� ��� �

, where�

is continuous for� , � , " � and periodic with period " � . The measure � � is

left-invariant (right invariant) since� ���� � � 4 � � � � � � � ���� � ��� � � � We now show how to construct a left-invariant measure for the � -dimensional

real linear Lie group � . Let 9 � � � be a parametrization of � in a neighborhood ofthe identity element and chosen such that 9 � ��� �

� . Set � � � � ��� 9 � � � � � �� � � , , � . Then by property (3) of a local linear Lie group, the � � � matrices $�� � 'are linearly independent. Given any other parametrization 9 � � � � near the identityand such that 9 ��� � � � � �

� we have 9 � � � � � 9 � � � � ��� for some analytic function� � � � and � �� � � � � 9 ��� � � � � � # � � �� 4 � �� 9 � � � � � �� � � � � � �

� � � � � � � �� 4 4 ' � ,� � � .Here the $�� �� ' must be linearly independent so ��� � 4 ' � ,� � �� �

. In other words, thetangent space at the identity element of � is � -dimensional and once we chose afixed basis $�� � ' for the tangent space, any coordinate system in a neighborhood of�� will determine � linearly independent � -tuples 4 ':4 , � . . . � 4 ' � , which generate

a parallelepiped with volume � � � ��� � 4 '" , � � � �in the tangent space. Now suppose 9 ��� � � is a parametrization of � in a neighbor-hood of the group element 9 �

and such that 9 ��� � � �&� 9 �. Then the matrices � �� �

� � � 9 � � � � � � � # � � � � . . . � � , are linearly independent. Furthermore, the matri-ces in a neighborhood of

��

( � can be represented as 9 � 4� 9 � � � � � 9 � � � � � � .Differentiating this expression with respect to � � and setting � � �

�we find9 � 4� � �� � �

� 4 4 ' � ,� � �

for � linearly independent � -tuples 4 ':4 , � . . .84 ' � , . This defines a parallelepiped inthe tangent space to � at 9 �

as the volume of its image in the tangent space at

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�� : � ' � ,� # � � ��� � 4 ' � ,� � � � �

By construction, our volume element is left-invariant. Indeed if � ( � then� � 9 ��� � 4 �

� � � � � 9 � � � � � � � � # � 9 � 4� � � 4 � � �� � 9 � 4� � �� � � � 4 4 ' � ,� � � �

so ��� � # � � � # . We define the measure � � 9 on � by

� � 9 � � � � � � � � 4 . . .+� � � (3.5)

Expression (3.5) actually makes sense independent of local coordinates. If � �� 4 � . . . � � � is another local coordinate system at 9 then9 � 4 � 9� � � � 9 � 4 � �� � � �� � �

� � : �� �

� � 4 ' � ,: � :so ��� �� � � � ��� �� � �

� �� � 4 ' � ,: �� � � � �

��� �

� � � �/. � � � � 4 ' � ,: � � Thus � � � � � � 4 . . . � � � � � � � � � ��� � � � � � � � � � 4 . . .7� �� � � � � � � � 4 . . .+� � � (3.6)

This shows that the integral�� � � 9 � � � 9 � �

� � �� 4 � . . . � � � � � � � � � � � 4 . . .7� � �

is well-defined, provided it converges. Furthermore,� � � � � 9 � � � 9 � � � � � � 9 � � � � � � � � � � � � � � � � � 9 � � � � ��� � � � � � �� � � � � 9 � � � � � � � � � � � � � 9 � � � 9 � (3.7)

where the third equality follows from the fact that � 9 runs over � if 9 does.By analogous procedures one can also define a right-invariant measure � ��9 in

� . Writing � � 9 � � � � 9 � 4 � � � � 0 ' � , � �

we define� � � � � � � � � � 0 ' � , � � � � ��9 � � � � � � � � 4 . . .+� � � (3.8)

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The reader can verify that � �89 is right-invariant on � .Since 9 � 9 � 4 � 9 � � � � 9 � 4 � � � 9 � � � � 9 � 4 , we have

� � � � � � � ��� )9!�/. � � � � � � (3.9)

where)9 is the automorphism � � 9 � 9 � 4 of the tangent space at the identity.

(That is, we consider)9 as an � � � matrix acting on the � -dimensional tangent

space.) Thus, if ��� )9 � � then � � 9 � �#�89 and there exists a two-sided invariantmeasure on � .

It can be shown that a much larger class of groups (the locally compact topo-logical (groups) possesses left-invariant (right-invariant) measures. Furthermore,the left-invariant (right-invariant) measure of a group is unique up to a constantfactor. That is, if

- 9 and-/� 9 are left-invariant measures on � then there exists a

constant � � � such that- 9 � � -/� 9 , [[43]], [[83]].

To illustrate our construction consider the Heisenberg matrix group � � , ex-ample (8) in � 2.1. Here ��� is a

�-dimensional Lie group which can be covered

by a single coordinate patch � � � � 4 � � � � � � � .9 � � � � ��� � � 4 � �� � � �� � �

� �Clearly, the matrices

� 4 � ��� � � �� � �� � �� � � � � � ��� � � �� � �� � �

� � � � � � ��� � � �� � �� � �� �

form a basis for the tangent space at the identity element of � � . Now

9 � 4 � 9 � ��� � � � 4 � � � � � 4 � � �� � � � �� � �� �� � 4 � � 4 � � � � � � 4 � � � � � � � � � � � �

where � 9 is the differential of 9 � � � . Thus,

� � � � �&� � � � ��� � � �� � � � 4� � �� ��� � �

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and � � 9 � � � 4 � � � � � � . Similarly,

� �-9 � 9 � 4 � ��� � � � 4 � � � � � 4 � � � �� � � � �� � �� �� � � 4 � � � � � � � � 4 � � � � � � � � � � � � �

so

� � � � � � � ��� ��� � � � � �� � �� � �� � � � �

and � �%9 � � � 4 � � � � � � .The affine group � � , example (7) of � 2.1 is " -dimensional and can again be

covered by a single coordinate patch:

9 � ��� � �&� �� �� � � � � � �

A basis for the tangent space at the identity is

� 4 �� � �� � � � � � �

� � �� � � �and 9 � 4 �-9 � � � � � � � �� � � � �

�� � 4 � � � � � � � ��

Therefore � � � ��� � � � � � �� 4� �� 4� � � � �

� �and � � 9 � � � � � � � . On the other hand,

� � 9 � 9 � 4 � � � � � � � � � � � � �� � �� 4� � � 4 � � � � � � � � � � � � Therefore

� � � � � � �&� � ����

4� ����� � � � � ��

and � �%9 � � � � � � . In this case the left- and right-invariant measures are distinct.

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3.3 Orthogonality relations for compact Lie groups

Now we are ready to extend the orthogonality relations to representations of alarger class of groups, the compact linear Lie groups. We say that a sequence of� � � complex matrices $/9 ' � , ' is a Cauchy sequence if each of the sequences ofmatrix elements $/9 ' � , ' � � ,�� � , � is Cauchy. Clearly, every Cauchy sequenceof matrices converges to a unique matrix 9 � � 9 � � 9 !� � � � ��� 9 ' � , . A set�

of � � � matrices is bounded if there exists a constant � � �such that� 9 �=, � for � , � � , � and all 9 ( �

. The set�

is closed provided everyCauchy sequence in

�converges to a matrix in

�.

Definition 9 A (global) group of � � � matrices is compact if it is a bounded,closed subset of the set of all � � � matrices.

As an example we show that the orthogonal group� � � � � � is compact. If9 ( � ��� � � � then 9 % 9 � � � , i.e.,

� � 4 9 � � 9 � �.- �

Setting ��

we obtain� � 9 �� � � � , so � 9 �=, � for all � � . Thus the matrix

elements of 9 are bounded. Let $/9 ' � , ' be a Cauchy sequence in� ��� � � � with

limit 9 . Then� � � � � � ��� � 9 ' � , � % 9 ' � , � 9 % 9 so 9 ( � ��� � � � and

� ��� � � � iscompact.

Now suppose � is a real, compact, linear Lie group of dimension � . It followsfrom the Heine-Borel Theorem [[96]] that the group manifold of � can be cov-ered by a finite number of bounded coordinate patches. Thus, for any continuousfunction

� � 9 � on � , the integral�� � � 9 � � � 9 � �

� � � 9 � � � � � � � �will converge (since the domain of integration is bounded.) In particular the inte-gral � � � �

� � � � 9 � (3.10)

called the volume of � , converges. The preceding remarks also hold for the right-invariant measure � �%9 . Moreover, we can show � � 9 � �#�89 for compact groups.

Theorem 7 If � is a compact linear Lie group then � � 9 � � ��9 .

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PROOF: By (3.9), � ��9 � � � � )9�� � � 9 , where)9 is the mapping � � 9 � 9 � 4 of

the tangent space at the identity of � . (We can think of)9 as an � � � � � matrix

rep of � .) Since � is compact the matrices 9 � 9 � 4 ( � are uniformly bounded.Thus the matrices

)9 are bounded and there exists a constant� � �

such that� � � )9�� , �for all 9 ( � . Now fix 9 and suppose � ��� )9�� � � �� � . Then

� � � )9 � � � � � � )9�� � � � � � � ��� � � � " � . . . Choosing appropriately we get � � � �

, which is impossible. Therefore � � �for all 9 ( � and � � 9 � �#��9 . Q.E.D.

For � compact we write � 9 � � � 9 � �#�%9 where the measure �-9 is both left-and right-invariant.

Using the invariant measure for compact groups we can mimic the proofs ofmost of the results for finite groups obtained in Chapter 2. In particular we canshow that any finite-dimensional rep of a compact group can be decomposed intoa direct sum of irred reps and can obtain orthogonality relations for the matrixelements.

For finite groups � these results were proved using the average of a functionover � . If

�is a function on � then the average of

�over � is

��� � � �� � � � ��� � � ��� � �� ��

If � ( � then��� � � � � � � � ��� � � � � � �&� ��� � � � � ��

(3.11)

Furthermore��� � � 4 � 4 � � � � � � � �� � � � � 4 ��� � � 4 � � � � � � ��� � � � � � � � ��� � � �&� � (3.12)

Properties (3.11) and (3.12) are sufficient to prove most of the fundamental resultson the reps of finite groups. Now let � be a compact linear Lie group and let

�be

a continuous function on � . We define

��� � � � 9 ����� �� � �� � � 9 � � 9 � �

� � � 9 ��- 9 (3.13)

where � 9 is the invariant measure on � , � � � � � � � 9 is the volume of � , and- 9 � � � 4� � 9 is the normalized invariant measure. Then��� � � � � 9 � �&� � � � � � 9 ��- 9 � � � � � 9 ��- 9 � ��� � � � 9 � � ���� � � � 9 � � �&� ��� � � � 9 � � � ��� � � � � � � - 9 � � � � ( � (3.14)

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since- 9 is both left- and right-invariant. Thus,

��� � � � 9 ��� also satisfies properties(3.11), (3.12).

In order to mimic the finite group constructions we need to limit ourselves tocontinuous reps of � , i.e., reps � such that the operators � � 9 � are continuousfunctions of the group parameters of 9 ( � .

Theorem 8 Let � be a continuous rep of the compact linear Lie group � on thefinite-dimensional inner product space � . Then � is equivalent to a unitary repon � .

PROOF: Let� . � . � be the inner product on � . We define another inner product� . � . � on � , with respect to which � is unitary. For � � � ( � define�

� � � � � �� � � � 9 � � � � � 9 ��� � - 9 � ��� � � � � 9 � � � � � 9 ��� � � (3.15)

(The integral converges since the integrand is continuous and the domain of in-tegration is finite.) It is straightforward to check that

� . � . � is an inner product.In particular, the positive definite property follows from the fact that the weightfunction is strictly positive (except possibly on a set of Lebesgue measure

�.) Now� � � � � � � � � � � � � � ��� � � � � 9 � � � � � � 9 � ��� � � � ��� � � � � 9 � � � � � 9 � � � � � �

� � � � �so � is unitary with respect to

� . � . � . The remainder of the proof is identical withthat of Theorem 1. Q.E.D.

Thus with no loss of generality, we can restrict ourselves to the study of unitaryreps.

Theorem 9 If � is a unitary rep of � on � and � is an invariant subspace of �then �

�is also an invariant subspace under � .

Theorem 10 Every finite-dimensional, continuous, unitary rep of a compact lin-ear Lie group can be decomposed into a direct sum of irred unitary reps.

The proofs of these theorems are identical with corresponding proofs for finitegroups.

Let $ � ' � , ' be a complete set of nonequivalent finite-dimensional unitary irredreps of � , labeled by the parameter � . (Here we consider only reps of � oncomplex vector spaces.) Initially we have no way of telling how many distinctvalues � can take. (It will turn out that � takes on a countably infinite number of

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values, so that we can choose � � � � " � . . . .) We introduce an���

basis in eachrep space � ' � , to obtain a unitary � � � � � matrix rep ' � , of � .

Now we mimic the construction of the orthogonality relations for finite groups.Given the matrix reps ' � , � ' � , , choose an arbitrary � � � � � matrix

�and form

the � � � � � matrix

# � ��� � ' � , � 9 � � ' � , � 9 � 4 � � � �� ' � , � 9 � � ' � , � 9 � 4 ��- 9

Just as in the corresponding construction for finite groups, one can easily verifythat ' � , � � � # � # ' � , � � �for all � ( � . Recall that the Schur lemmas are valid for finite-dimensional repsof all groups, not just finite groups. Thus if � �� �

, i.e., ' � , not equivalent to ' � , , then # is the zero matrix. If � � �then # � � �

� � for some� ( �

:

# � � � � � � � � � � � � � ��- � �����

Letting�

run over all � � � � � matrices, we obtain the independent identities�� ' � , � � 9 � ' � , : � 9 � 4 ��- 9 � � � � ��� � ��- � �

- : � (3.16)

for the matrix elements ' � , � � 9 � . To evaluate�

we set��� � � � � � and sum on � :

� �

4 � � � �� � �

�� �

! 4 '� , � 9 � 4 � ' � , � � 9 ��- 9 � - �

Therefore� �.- � � � . Since the matrices ' � , � 9 � are unitary, (3.16) becomes�

� ' � , � � 9 � ' � ,: � 9 ��- 9 � � - :� � ���- � - � �

� � ,�� ��� , � �� � , � � , � �

(3.17)

These are the orthogonality relations for matrix elements of irred reps of � .

3.4 The Peter-Weyl theorem

In the case of finite groups � we were able to relate the orthogonality relations toan inner product on the group ring

�� . We can consider

�� as the space of all

functions� �� �

on � . Then� � 4 � � � � � ��� � � ��� � 4 �� � � � � �

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defines an inner product on�� with respect to which the functions $ � 4 8 �� ' � , � � � � 'form an

� �basis. We extend this idea to a compact linear Lie group � as fol-

lows: Let* � � � � be the space of all functions on � which are (Lebesgue) square-

integrable: * � � � � � $ � � 9 � % � � � � � 9 � � � - 9���� ' (3.18)

With respect to the inner product� � 4 � � � � � �� � 4 � 9 � � � � 9 ��- 9 � (3.19)* � � � � is a Hilbert space, [[43]], [[83]]. Note that every continuous function on �

belongs to* � � � � . Let

� ' � , � � 9 �&� � 4 8 �� ' � , � � 9 �� (3.20)

It follows from (3.17) and (3.19) that $ � ' � , � ' , where � , � � , � � and � rangesover all equivalence classes of irred reps, forms an

���set in

* � � � � .For finite groups we know that the set $ � ' � , � ' is an

���basis for the group ring,

and every function�

on the group can be written as a unique linear combinationof these basis functions. Similarly one can show that for � compact the set $ � ' � , � 'is an

� �basis for

* � � � � . Thus, every� ( * � � � � can be expanded uniquely in

the (generalized) Fourier series

� � 9 � � �

� �

� 4 �� � ' � , � 9 � (3.21)

where � � � � � � � ' � , � (3.22)

Furthermore we have the Plancherel equality� � 4 � � � � � �

� �

� 4 �� '54 , � � '2� ,

(For� 4 0 � � this is called Parseval’s equality.) Convergence of the right-hand

side of (3.21) to the left-hand side is meant in the sense of the Hilbert space norm.We will not here take up the question of pointwise convergence. See, however,[[38]], [[39]], [[63]], [[92]].

Theorem 11 (Peter-Weyl). If � is a compact linear Lie group, the set $ � ' � , � ' isan� �

basis for* � � � � .

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The proof of this theorem depends heavily on facts about symmetric com-pletely continuous operators in Hilbert space and will not be given here. For thedetails see [[43]] or [[83]].

Corollary 3 A compact linear Lie group � has a countably infinite (not finite)number of equivalence classes of irred reps $ � ' � , ' . Thus, we can label the repsso that � � � � " � . . . .PROOF: The functions $ � ' � ,� ' form an

� �basis for

* � � � � . Since* � � � � is a

separable, infinite-dimensional Hilbert space there are a countably infinite numberof basis vectors [[43]], [[83]]. Q.E.D.

We illustrate the Peter-Weyl theorem for an important example, the circlegroup

� � � � , example 5 in � 2.1. Since� � � � � $ � ���& �� ' , � � � � � � � , is compact

and Abelian its irred matrix reps are continuous functions �� � �

such that

� � � 4 � � � �&� � ��� 4 � � � � � � � � 4 � � � ( � � (3.23)

and � ��� � � � � � ��� � . The functional equation (3.23) has only the solutions� � � � � � � � and the periodicity of � implies �

� " � � � , where � is an integer.Therefore, there are an infinite number of irreducible unitary representations of� � � � :

� ���� �&� � ���� � � � � � � � � � � � " � . . .

The normalized invariant measure on� � � � is � � . The space

* � � � � � � � is just thespace

* � � � � � � � � consisting of all measurable functions� � � �

with period � suchthat

� 4� � � ��� � � � � � ��� . By the Peter-Weyl theorem the functions $ � ���& � � ' forman� �

basis for* � � � � � � � � . Every

� ( * � � � � � � � � can be expressed uniquely in theform

� ��� ��

� � � � � ���� � � � � � � � 4� � � � � � � ���� � � � �0 (3.24)

Furthermore, � 4� � � ��� � � �7� � � � � � � � � � (3.25)

Here (3.24) is the well-known Fourier series expansion of a periodic function and(3.25) is Parseval’s equality.

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3.5 The rotation group and spherical harmonics

A second very important example of the orthogonality relations for compact linearLie groups and the Peter-Weyl theorem in the rotation group

?� � � � � ?

� � � � � � .Here we will give some basic facts about the irreducible representations of

?� � � �

and refer to the literature for most of the proofs, [[20]], [[45]], [[79]], [[107]].Recall that

?� ��� �

has a convenient realization as the group of all� � � real

matrices 9 such that 9 % 9 � � � and ��� 9 � � , example 5, � 2.1. This is thenatural realization of

?� ���#�

as the group of all rotations in� � which leave the

origin fixed. One convenient parametrization of

?� � � �

is in terms of the Eulerangles. Recall that a rotation through angle � about the � axis is given by

��� � � ��� ��� � � � � � ��� � � �� ��� � � � � � �� � �

�� ( ?� ��� �

and rotations through angle � about the � and � axis are given by

� � � � � � ��� � � �� � � � � � � ��� �� ��� � � � � � �� � ( ?

� ��� � �� � � � � � ��� � � � � � � ��� �� � �

� � ��� � � � � � ��� ( ?

� ��� � �respectively. Differentiating each of these curves in

?� � � �

with respect to � andsetting � � �

we find the following linearly independent matrices in the tangentspace at the identity:

� � � ��� � � � �� � �� � �

� � � � � � ��� � � �� � � �� � �� � � � � � ��� � � �� � �

� � � �� � (3.26)

One can check from the definition 9 % 9 � � � that the tangent space at the identityis at most three-dimensional, so the matrices (3.26) form a basis for this space.

The Euler angles � � � � for 9 ( ?� ��� �

are given by9 � � � � � � � ��� � � � � � � ����� � � � � (3.27)

��� � � � � � � � � � � � � � ��� � �� � � �

� � � � ��� � � � ��� � � � � � � � � � � � � � � ��� ��� � � � � � � � � � � � ��� � �� � � � � � � � ��� � � �

� � � � � � � � � � � �� � � � � � � ��� � � ��� � � � � � � ��� � � �

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It can be shown that every 9 ( ?� ��� �

can be represented in the form (3.27) wherethe Euler angles run over the domain� , � �" � � � , 1, � � � , �

��" � (3.28)

The representation of 9 by Euler angles is unique except for the cases � � � �where only the sum � � �

is determined by 9 , but this exceptional set is onlyone-dimensional and doesn’t contribute to an integral over the group manifold.The invariant measure on

?� � � �

can be computed directly from the formulas of� 3.2. Let 9 � � � � � �)( ?

� � � �. Then9 � 4 � ���� � � ��� � � ��� � � � � � � � ��� � � � � � � � � � �9 � 4 � ��

�� � � � � � � � ��� � � � � �9 � 4 � ���� � � ��

Thus,

� � � � � � � � � � ��� ��� ��� � � ��� � � � � � � � � � � � � � � � � ��� � � �� � �� � � � � ���

and � 9 � ��� � � � � �� � (3.29)

Since

?� � � �

is compact, � 9 is both left- and right-invariant. The volume of?� � � �

is

����� ' � , � ���� ' � , � 9 � � ���� � � � ���� � � � �� � ��� � � � � � (3.30)

The irred unitary reps of

?� ��� �

are denoted � ' � , ��� � � � � � " � . . . , where ���� � ' � , �" � � � . (In particular ' � , � 9 � � � and '54 , � 9 � � 9 .) Expressed in terms of an���basis for the rep space � ' � , consisting of simultaneous eigenfunctions for the

operators � ' � , � ��� � � ��� , the matrix elements are

� � � � � � � � � � � � � ' � � � , ' � � ,' � � , ' � � � , � 4 8 � �� (' � � � � , ) � ��� � *������ ':4 ����� � � , � ������ �� ' � � � 4 , � � 4�

� � � � � �� � � � � ��� � �

� 4��� � �� 4 �� � � � � ' � � � , ' � � ,' � � , ' � � � , � 4 8 � � 5' � � � � ,� � � �� � � � � � �

� � , � � , � (3.31)

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Here � � 4�� �� � � � is the Gaussian hypergeometric function and � � � � is the

gamma function [[40]], [[107]], [[110]]. A generating function for the matrixelements is

�� 9 � � � � � 0 � ���4 � � � � � 4 � � �0 � � � �� � � � � � � � � � � � � � 4 8 � � �

� � ��� 9 � � � � � � � �

� � � � � � � � � ��� � � � 4 8 �

where 4 � � (' � � � ,:8 � � � � " � 0 � � � (' � � � ,:8 � ��� � " The group property

� � � 9 4 9 � �&�� � � �

� � � 9 4 � �� � � 9 � �defines an addition theorem obeyed by the matrix elements. The unitary propertyof the operator � ' � , � 9 � implies

� � � 9 � 4 � � �� � 9 � �or in Euler angles,� ��� � � � � � �� � � � � ��� � � � � � � � � � � �� � � � � � � � � � � � ��� � � � � � �� Also, � � � � 9 � �-, � or

� � � �� � � � � � � , � � � � � � � � � � � �� � � � � � � � � � ��� 4 8 � � � , 1, � The matrix elements �� � � � � � � � , are proportional to the spherical harmonics� �� � � � � . Indeed

�� � � � � � � � � � � ) � �" � � � / 4 8 � � �� � � � � � � � � � � � � � �� ��� � � � � 4 8 � �� � � � � � � � �

where the �� � � � � � are the associated Legendre functions [[40]], [[45]], [[78]],[[79]]. Moreover, �� � � � � � � � � � � � � � �

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where � � � � � � is the Legendre polynomial.According to the general theory of � 3.3, the matrix elements � � � 9 � satisfy

the orthogonality relations���� ' � ,

��� 9 � � � � 9 � � 9 � � � �" ��� � - - � �� - � � (3.32)

Thus� ���� � � � ���� � � � �� � � � � � � � � � � � � � � � � � ��� � � � � �" � � � - � - � � - � � The

�and � integrations are trivial, while the integration gives� ��

� �� � � � � � � �� � � � � � ��� � �� � "" ��� �� � � � � � � � � � �� � � � � � � � � � � - � �

For!� � � �

these are the orthogonality relations for the Legendre polynomials.(Note: By definition,

� � � �� � � � � � � �� � � � � � � � � �� � � � � �,� � � � � � � , where �� � � are Legendre functions.)

By the Peter-Weyl theorem, the functions

� � �� � � � � � � � " � � � � 4 8 � � � � � � � � � �

� � , � � , � � � � � � � � " � . . .constitute an

� �basis for

* � �?� � � � �

. If� ( * � �

?� ��� ���

then

� � � � � � � � � �

� � � � � �

� �� �

�� � � � � � (3.33)

where� � �

� � � � � � �� � 4� � � ���� � � � ���� � � � �� � �

� � � � � � � � � � � � � � � � � � ��� (3.34)

Some particular cases of (3.33) are of special interest. Suppose� � � � �)( * � �

?� ��� ���

is independent of the variable � . If we think of� � � � as latitude and longitude,

we can consider�

as a function on the unit sphere

?� � �

?� ���#� � � � � , square-

integrable with respect to the area measure on

?� . Since the � -dependence of� �

�� � � � � � is � � , it follows from (3.34) that �

� �� �

unless � �

; the onlypossible nonzero coefficients are �

�� � . Now

� �� �� � � � � � � ��� � � 4 8 � � �� � � � �

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where� �� is a spherical harmonic. Thus,

� � � � �&� � �

� � �

��� �� � � � � (3.35)

where

� � � � � ���� � � � �� � � � � � � � �� � � � � ��� � � � � �� � � � � �&�.- � � - � �� This is the expansion of a function on the sphere as a linear combination of spheri-cal harmonics. Again, (3.35) converges in the norm of

* � �?� � � � �

, not necessarilypointwise.

If� � � ( * � �

?� ���#� �

is a function of alone then the coefficients �� � are zero

unless�� � � �

. Here,

� �� �

� � � � � � � � " � � � � 4 8 � � � � � � � � � � � � � � " � . . .where

� � � �&� � � 4���� � � �� � � �" � � ) � � �" / � � � 4

�� � � �� � � �

� � � �is a Legendre polynomial of order � . The coefficient of �

�in the expansion of

� � � � is nonzero and � � � � � � . The expansion of� � � becomes

� � � � � � �� � � � � � � � � �� � � 4� � " � � � � � �� � � � � � � � � � ��� � �� �

� �� � � � � � � #� � � � � � ��� � � � �

� � � � 4 (3.36)

3.6 Fourier transforms and their relation to Fourierseries

Abelian groups � , (not necessarily compact) are another class of groups concern-ing which one can make general statements about the decomposition of

* � � � � interms of unitary representations [[20]], [[39]], [[43]], [[63]], [[83]], [[92]], [[107]].We will not go into this theory but consider only a single, very important, examplewhere the results are familiar to everyone: The group

�of real numbers

�with

addition of numbers as group multiplication. Here�

is isomorphic to the matrixgroup

�;�, examples (1) and (3), Section � 2.1, and � � is the invariant measure. The

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unitary irred reps of�

are one-dimensional, hence continuous functions� �����

suchthat � ��� 4 � � � � �.� ��� 4 � � ��� � � � � 4 � � � ( � (3.37)

This functional equation has only the solutions� ������� � � % and the unitarity re-

quirement implies �� " � � @ where

@is real. Given a function � ( * � � � � we

have � �����&� � � ? �A@ � � ���& 2! % � @

where the Fourier coefficients

? �A@ �are defined by? �3@ � � � � � ����� � � ���& 2! % � �� � � � � ! � �

and� ! ������� � ���� 2! % is the irred rep. Parseval’s equality is� � � � ����� � � � � � � � �

? �A@ � � �+� @ Formally, the orthogonality relations are� � ! � � ! �&� � � ��� � " � � � �3@ � @ � � � � � �.- �3@ � @ � �where

- �A@ �is the Dirac delta function. Note that the sum over irred reps, familiar

for compact groups, is here replaced by an integral over irred reps.It is illuminating to compare the Fourier transform on

�with the correspond-

ing results for� � � � and � � . Assume that

� ( * � � � � belongs to the Schwartzclass, i.e.,

�is in

� � � �and there exist constants

�� � � (depending on

�) such

that � � ������ % � � �-, �� � � on

�for each � � � � � � � � " � . . . . Then the projection operator

maps�

to a continuous function in* � � � � � � � � with period one:

� � � � � ��� � � � � � � � � (3.38)

Expanding � � � � � � into a Fourier series we find

� � � � � � � � � � � � ���� � �

where � � � � 4� � � � � � � � � ���& � � � � � ��� � � � � � � ���� � � � � � �� � � �52

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and �� �A@ � is the Fourier transform of� � � � . Thus,

� � � � � � ��&�

� � �� � � � � ���& � � � (3.39)

and we see that � � � � � � tells us the value of �� at the integer points@ �

� , butnot in general at the non-integer points. (For �

� �, equation (3.39) is known as

the Poisson summation formula, [[39]]. If we think of�

as a signal, we see thatperiodization (3.38) of

�results in a loss of information. However, if

�vanishes

outside of � � � � � then � � � � � � 0 � � � � for� , � � � and

� � � ��� ��� � � � � ���� � � � � , � � � (3.40)

without error. Now suppose (3.40) holds, so that there is no periodization error.For an integer

� � � we sample the signal at the points � � � � � � � � � . . . � � � � :� ) �� / �

��� � � � � ���& � � 8 � � � , � �

� (3.41)

From the Euclidean algorithm we have �� � � � � where

� , � ��

and � � � areintegers. Thus

� ) �� / � � � 4

� �� � �� � � � � � � � � ���& � � 8 � � � , � �

� (3.42)

Note that the quantity in brackets is the projection of �� at integer points to aperiodic function of period

�. Furthermore, the expansion (3.42) is essentially

the finite Fourier expansion (2.37). However, simply sampling the signal at thepoints � � tells us only

� � �� � � � � � � not (in general) �� � � � . This is known asaliasing error.

Although we will not work out the details in these notes, a similar approachto the foregoing (with periodizing and aliasing error) is appropriate and useful forFourier analysis on the Heisenberg and affine groups.

3.7 Exercises

1. Construct a real irred two-dimensional rep of the circle group��� � � .

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2. Show how to decompose any real finite-dimensional rep of� � � � as a direct

sum of real irred reps.

3. Verify that the measure � �89 , (4.8), is right-invariant on the group � .

4. Prove: If � is a compact linear Lie group then � � 9 � 4 �&� � 9 , i.e.,� � � � 9 � 4 � � 9 �

� � � � � � �-9 . Hint: Show that � � � � � � �*� � � � � � )9 � � � � � � � where)9 is the

automorphism� � 9 � 9 � 4 of ��� � matrices

�.

5. Assuming that� �����

is a continuously differentiable function of�, use a dif-

ferential equations argument to show that the only nonzero solutions of thefunctional equation� ��� 4 � � � �&��� ��� 4 � � ��� � � � � 4 � � � ( �are

� ������� � � % , where � is a constant.

6. Assuming only that that� �����

is a real continuous function of�, show that

the only nonzero solutions of the functional equation� ��� 4 � � � �&��� ��� 4 � � ��� � � � � 4 � � � ( �are

� ����� � � � % , where � is a constant. Hint: Set� ����� � � �$'"% , so that

� ��� 4 �� � ��� � ��� 4 � � � ��� � � . Then determine� �����

for�

rational from� � � � .

7. Use the Poisson summation formula for the Gauss kernel� � � ��� � " � ��� � � � ��� ��

to derive the identity

�� " � �

� � ���� �� �

� � �� ��� � %

Hint: �� �A@ � � � � ��� ! % .

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Chapter 4

REPRESENTATIONS OF THEHEISENBERG GROUP

4.1 Induced representations of� �

Recall that the Heisenberg group ��� can be realized as the linear Lie group ofmatrices 9 � � � � ��� � � 4 � �� � � �� � �

� � (4.1)

with group product� � 4 � � � � � � � . � � 4 � � � � � � ��� � � 4 � � 4 � � � � � � � � � � � � � � 4 � � � �and invariant measure � � 9 � �#��9 � � � 4 � � � � � �

To motivate the construction of the unitary irred reps of � � we review theessentials of the Frobenius construction of induced representations. Let � be alinear Lie group and � a linear Lie subgroup of � . If � is a rep of � on thevector space � we can obtain a rep ��� of � by restricting � to � ,

��� � � � � � � � � � � ( � On the other hand the method of Frobenius allows one to construct a rep of � froma rep of � . Let � be a finite-dimensional unitary rep of � on the inner productspace � . Denote by

� �the vector space of all functions � � 9 � with domain � and

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range contained in � where addition and scalar multiplication of functions are thevector operations. Here, for a fixed 9 ( � , � � 9 � is a vector in � . Let � � be thesubspace of

� �defined by

� � � $ � ( � � % � � � 9 � � � � � � � � 9 � � � � ��� � � (� � 9 ( � ' (4.2)

We define a rep � � of � on � � by

� � � � 9 � � � � 9 � �&� � � 9 � 9 � � 9 � 9 � ( � � � ( � � (4.3)

It is clear that � � is invariant under � and the operators � � � 9 � satisfy the ho-momorphism property. Here, � � is called an induced representation. If

� . � . � isthe inner product on � and � is compact we can initially define the inner product� . � .3� on � � by �

� 4 � � � � � �� � � 4 � 9 � � � � � 9 � � � �%9

where �#�89 is the right-invariant measure on � . Then we restrict the operators � �to the subspace � � � � � of functions � such that

�� � �/� � � . If 9 � ( � and

� ( � � we have� � � � 9 � � � 4 � � � � 9 � � � � � � � � � � 4 � 9;9 � � � � � � 9 9 � � � �#��9� � � � � 4 � 9 � � � � � 9 � � �#��9 � �� 4 � � � �

so � � is unitary. However, note from (4.2) that� � 4 � � 9 � � � � � � 9 � �&� � � 4 � 9 � � � � � 9 � �

for all � ( � , i.e., the inner product� . � . � is constant on the right cosets � 9 . Thus

if � is noncompact the above integral will be undefined (or � � will contain onlythe zero function). If the coset space � � ��� � admits a � -invariant measure,i.e., a measure � � � � � such that � � � � 9 � � � � � � � for all �

( � � 9 ( � , then wecan define the inner product on � � as�

� 4 � � � � � ��� � 4 � � � � � � � � ��� � � � � � (4.4)

and the operators � � % � � � � � � � � � � � � 9 � will still be unitary. Here, we choosean 9 �� ( � in each right coset � 9 � � � , so that 9 �� 9 � � 9 �� � and � � � � 0 � � 9 �� � .(Note: It is always possible to find local coordinates � � �

� 4 � . . . � � � � on � suchthat � � �

� 4 � . . . � ��� � , � , � , are local coordinates on � and � � �� � � 4 . . . � � � �

are local coordinates on � .) In general, no such invariant measure � � exists, butit does exist in many cases. In particular, if both � and � are unimodular, i.e.,if the left-invariant and right-invariant measures for each of these groups are the

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same, and � is a closed subgroup of � then it can be shown that an (unique up toa constant multiplier) invariant measure � � exists and that in local coordinates

�-9 � � � � � � � � 4 . . .+� � � � 9 ( �� � � � � � � � � � 4 . . .7� ��� � � (�

and � � � � � � � � � � � � ��� � 4 . . .+� � �where � 9 � � � � � . Thus � � � � �&� � � � � � � � � � .4.2 The Schrodinger representation

Let us consider the case where � � ��� and � is the subgroup of matrices$/9 � � � � � � � � � ' . Since 9 � � � � � � � � � 9 � � � � � � � � � � �&� 9 � � � � � � � � � � � � � � � � � it is clearthat � is Abelian and that the operators� � �!9 � � � � � � � � � � � � ���& � ��� define a one-dimensional unitary irred rep of � . Fur-thermore the left-invariant and right-invariant measure on � is � � � � � � . Since9 � � 4 � � � � � � � � 9 � � � � � � � � � 9 � � 4 � � � � �it is clear that the coset space ��� � can be parametrized by the coordinate � 4 ��

,and the (scalar) functions in � � can be taken as

� �����. If 9 � � 4 � � � � � � � acts on this

function, it transforms to

� ��� � � 4 � � � � � � � � � � � 0 � � � � � � ��� � � 4 �where � � 9 � � � � � � � � � � � � � , (4.2). Thus the action of ��� restricted to the cosetspace is

� � � 9 � � ��� � ����� 0 � � � � 4 � � � � � � � � ����� � � ���& � ' ��� � % � , � ��� � � 4 �� (4.5)

One can verify directly that � � is a rep, i.e.,

� � � 9 4 � � � � 9 � � � � � � 9 4 9 � �and that it is unitary with respect to the inner product� � 4 � � � � � � � � 4 ����� � � ����� � �� (4.6)

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For� � �

this rep is reducible but (as we will show later) for� �� �

it is irred in thesense that there is no nontrivial closed subspace of

* � � � � which is invariant underthe operators � � � 9 � � 9 (

� � . This rep is called the Schrodinger representationof the Heisenberg group.

Note that the mapping 9 � � 4 � � � � � � ���� � � 4 � � � �is a homomorphism of ��� onto the Abelian group

� � % � � 4 � � � � . � � 4 � � � ��� � � 4 �� 4 � � � � � � � . The unitary irred reps of� � are clearly of the form

� 2 � � 2 � � 4 � � � �&�� ���& (' 2 � � � � 2 � , for real constants 4 4 � 4 � . It follows that the matrices 2 � � 2 � 9 � � ������ ���& (' 2 � � � � 2 � , define one-dimensional unitary irred reps of � � . It can be shownthat � � and 2 � � 2 are the only irred unitary reps of ��� , [[98]].

4.3 Square integrable representations

Assuming for the time being that � � is irred for real� �� �

let us see if thereis an analog for � � of the orthogonality and completeness relations for matrixelements which hold for linear compact Lie groups, and for compact topologicalgroups in general. Noncompact groups � can have infinite dimensional irredunitary reps. That is, the rep space is a separable infinite dimensional Hilbertspace

�with inner product

� . � . � . The rep operators � � � � � � ( � are unitaryi.e., each � � � � is a linear mapping of

�onto itself which preserves inner product:� � � � � � 4 � � � � � � � � � � � 4 � � � � , for all

� 4 � � � (��. The rep is irreducible if there is

no proper closed subspace of � which is invariant under the operator � � � � � � ( � .The reps � � � � of � on the Hilbert spaces

� � � �, respectively, are equivalent if

there is a bounded invertible linear operator� %�� � � �

such that� � � � � �� � � � � � for all �

( � , i.e.,� � � � � � � 4 � � � � � � .

With these definitions the analog of Theorems 2, 4 are true and can be provenwith ease. Moreover, the following analog of Theorem 5 holds:

Theorem 12 Let � be a unitary rep of the group � on the separable Hilbert space�

. Then � is irred if and only if the only bounded operator�

on�

satisfying

� � � � � ��� � � � � (4.7)

is� � � � where � is the identity operator on

�.

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SKETCH OF PROOF: Part of this result is easy to prove. Suppose� � � � is

the only solution of (4.7) and let � be a closed subspace of�

which is invariantunder � % � � � � � ( � for all �

( � � � ( � . Then ��

is also invariant under� and� � � ���

�. Thus for every � (��

we have the unique decomposition� � � 4 � � � � � 4 ( � � � � ( � �

where � � � � � 4 ( � and � � � � � � ( � �. Now

define the projection operator � by � � � � 4 . Clearly � is a bounded operatoron�

and � � � � � � � � � � � for all �( � . By hypothesis, � � � � . If

� � �then � is the zero subspace; if

� �� �then

� � � and � � �. Thus � is not a

proper invariant subspace and � is irred.The converse is somewhat more difficult. Suppose � is irred and

�is a

bounded linear operator satisfying (4.7). Then the adjoint���

of�

also satisfies(4.7) so, without loss of generality, one can assume that

�is self-adjoint. Then,

by the spectral theorem for self-adjoint operators [[1]], [[2]], [[43]], [[83]], [[95]],the projection operators � � in the spectral family associated with

�must all com-

mute with each � � � � . By hypothesis then, each � � is either the zero operator orthe identity operator. Hence

� � � � for some real number�

. Q.E.D.Note: At this point it is appropriate to mention that if the unitary irred reps� � � � of a group � are equivalent, then they are unitary equivalent, i.e., there

is a unitary operator � such that � � � � �$� � ��� � � � for all �( � . (Here �

maps the Hilbert space�

onto the Hilbert space� �

and preserves inner product:�� � 4 � � � � � � � �

� 4 � � � � for all � 4 � � � ( �. In particular, this means that �

� ��� 4

where the adjoint��� % � � � �

of a bounded operator� % � � � �

is de-fined by

� � � � � � � � � �� � ��� � � � for all

� ( � � � � ( � �� ) Indeed, if � � � �

then there is a nonzero bounded invertible operator� % � � � �

such that� � � � �,� � � � ��� . Taking the adjoint of both sides of this equation and using thefact that � � � � � � � � 4 � � � � � � � � �*� � � 4 � � � we have

��� � � 4 � � �*� � � 4 � � �?�.

Eliminating � � � � � from the two equations we find� � � � � � � 4 � � � � 4 � � � ��� � or� � � � � � � � � � � � � �'� � � �0� . Since � is irred it follows from Theorem 12 that� � ��� 4 � where � is the identity operator and 4 is a constant. Since 4 � � � � � � �� ��� � � � �/� � � � � � � � � � � � � � � � � � �

for � �� , then 4 � 0 � is a positive con-stant. Setting �

� 0 � 4 � we have ���� 4 � 4 ��� � � � , so � is unitary and

� � � � � � � � � � � � .Based on these results we can mimic the proof of the orthogonality relations

(3.17) for any linear Lie group which is unimodular, i.e., such that � ��9 � � � 9 .There are two differences, however: (1) In order that the integrals (3.17) exist wemust limit ourselves to those irred reps � ' � , of � whose matrix elements ' � ,� � 9 �are square integrable. (2) We cannot normalize the measure in general, since the

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volume of � may be infinite. Thus we obtain the result�� ' � , � � 9 � ' � ,: � 9 � � 9 � - :� � � � - � - � � (4.8)

where ' � , � � 9 � are the matrix elements of � ' � , with respect to some� �

basis and� ' � , � � ' � , are unitary irred reps of � whose matrix elements are square integrable.The constant � � � � � �

is called the degree of � ' � , . (It can be shown that ifone matrix element ' � , � � 9 � of � ' � , is square integrable, then all possible matrixelements

� � ' � , � 9 � � ��� � are square integrable.) Furthermore, (4.8) can be writtenin a basis-free form analogous to (2.24).

Theorem 13 For� �� �

the representation � � of � � on* � � � � is irreducible.

PROOF: We will present the basic ideas of the proof, omitting some of the tech-nical details. Our aim will be to show that if

�is a bounded operator on

* � � � �which commutes with the operators � � � 9 � for some

� �� �and all 9 ( � � then� ��� � for some constant

�, where � is the identity operator. (It follows from

this that� �

is irred, for if � were a proper closed subspace of* � � � � , invariant

under � � , then the self-adjoint projection operator � on � would commute withthe operators � � � 9 � . This is impossible since � could not be a scalar multiple of� .)

Suppose the bounded operator�

satisfies

� � � � � 4 � � � � � � � � � � � � 4 � � � � � � � �for all real � . First consider the case � 4 � � � � �

:�

commutes with theoperation of multiplication by functions of the form � � � % for real � . Clearly

must also commute with multiplication by finite sums of the form��� � � � ���� � � � %

and, by using the well-known fact that trigonometric polynomials are dense inthe space of measurable functions,

�must commute with multiplication by any

bounded function� �����

on� � � � � � . Now let

�be a bounded closed internal in� � � � � � and let

��� ( * � � � � be the characteristic function of�

:

���&�����&� � � ��� � ( �� � � � ( � Let

� � ( * � � � � be the function� ��� � ���

. Since� � � � ���

we have� �&�����*�

� ���&����� � � � � � ����� � ���&�����%* ���&����� � ���&����� � � �����so

� �is nonzero only for

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� ( �. Furthermore, if

� �is a closed interval with

� � � � and� � � � ��� then

� � ������� � ��� ��� �����&�.��� ������� ���&�����&����� ����� � � ����� so� � �����&� � �&�����

for�)( � �

and� � ����� � �

for� ( � � . It follows that there is a unique function

� �����such that���� � ( * � � � � and

���� ����� � ����� � � ���� �����for any closed bounded interval

)�in� � � � � � . Now let � be a

� function which is zero in the exterior of

)�. Then� � ������� � � � ���� �������&� � ����� � ���� ������� � ����� � ����� ���� ������� � ����� � ����� , so

�acts on �

by multiplication by the function� �����

. Since as)�

runs over all finite subintervalsof� � � � � � the functions � are dense in

* � � � � , it follows that�%� � ����� � .

Now we use the hypothesis that� � � � � 4 � � � � � � ����� � � � � � 4 � � � � � � � ����� for all

� 4 and � (+* � � � � % � ����� � ��� � � 4 ��� � ��� � � 4 � � ��� � � 4 � . Thus� ������� � ��� � � 4 �

almost everywhere, which implies that� �����

is a constant. Q.E.D.

4.4 Orthogonality of radar cross-ambiguity functions

The results of � 4.3 do not directly apply to the unitary rep � � of � � becausethis rep fails to be square integrable. Indeed it is evident from (4.5) that the � � -dependence of the matrix element

� � � � � � � 4 � � � � is of the form � ���& � ��� , so theintegral � � � � � � � � � � 4 � � � � � � � � � � � � � � < �7� � 4 � � � � � �

will diverge. All is not lost because we can factor out the center of � � andconsider only the factor space. The center � consists of all elements of � � whichcommute with every element of ��� . Clearly

� � $/9 � � � � � � � �&� 9 � � � � � � � ( � ' Now � � is irred and � � � � � � � � � � �&� � � � � � � � � � � � for all 9 � � � . From Theorem12, � � � � � � must be a multiple of the identity operator on

* � � � � . Indeed we knowthat � � � � � �&� � ���& � ��� � . Now9 � � �&� 9 � � 4 � � � � � � 9 � � � � �and since � � is irred, the unitary operators � � � � 4 � � � � � � , � 4 � � � ( �

, must actirreducibly on

* � � � � . Furthermore the measure � � 4 � � � is (two-sided) invariantunder the action of ��� . Thus we can repeat the arguments leading to (4.8) for reps� ' � , � � ' � , 0 � � and measure � � 4 � � � to obtain (with the assumption, correct aswe shall see, that the matrix elements �� � 9 � are square integrable):�� ��� �� � � � 4 � � � � � � �: � � 4 � � � � � � � � 4 � � � � - � : - � � � � � (4.9)

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In fact, the structure of � � is so simple that one can use ordinary Fourieranalysis to evaluate the left-hand side of (4.9) for arbitrary matrix elements. Theresult is� � � � � � 4 � � � � � � � � 4 � � � � � 4 � � � � � � < � � � � � 4 � � � � � � � � � � 4 � � � � � � < � � � � (4.10)

where � � � � � 4 � � � � � 4 � � � � � � � � ���� � % � � 4 ��� � � 4 � � � ����� � �� (This shows explicitly that the matrix elements

� � � � � 4 � � � � � 4 � � � � are square in-tegrable.

At this point we recall that the narrow-band cross-ambiguity function can bewritten in the form

�� �� � � 4 � � � #" � � ���� � � � ��� �

���� � � 4 � � � ����� � ���� ;% � � � (4.11)

so that the cross-ambiguity function differs from the matrix element� � 4 � � 4 � � � � � � � � � � �)� of � � by the simple multiplicative factor � �& � � � . Thus theresults of group representation theory can be brought to bear on the radar ambi-guity function. (For the purposes of computation of the ambiguity function,the phase factor � �& � � � is of no concern and we henceforth will identify thematrix element itself with the ambiguity function.)

Note the special case of (4.10) where� � 0 �

and � � � � � � � :� � � � � � 4 � � � � � � � � � 4 � � � � � � � � � � � � (4.12)

For� � � this is the radar uncertainty relation. (The maximum of � �� � � � 4 � � � � � � � � � is � and occurs for � 4 � � � � �

. However, in view of (4.12)the graph of this function cannot be too “peaked” around the maximum.)

From (4.10) we see that if $ � � � � � � � � � " . . . ' is an� �

basis for* � � � � then

the matrix elements $ � � � � � 4 � � � � � � � � �)� ' form an���

set in* � � � � � . This set

is actually an� �

basis for* � � � � � , see Exercise 5.2. This allows us to expand

a moving target distribution function in the narrow-band case as a series inthis ON basis, [[113]].

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4.5 The Heisenberg commutation relations

To motivate our next example we make a brief digression to study the Heisenbergcommutation relations of quantum mechanics. Recall that the matrices

� 4 � ��� � � �� � �� � ��� � � � � ��� � � �� � �� � �

�� � � � � ��� � � �� � �� � ���

form a basis for the tangent space at the identity element of � � . Defining thecommutator [ . � . ] of two

� � � matrices 9 and � by �!9 ��� � � 9 � � � 9 , we seethat � � 4 � � � � � � � � � � 4 � � � � ��� � � � � � � � � ��� (4.13)

where�

is the zero matrix. (Indeed, one can show that the tangent space at theidentity for any local linear Lie group � is closed under the commutator operation:if 9 and � belong to the tangent space, then so does �!9 ��� � . The tangent spaceequipped with the commutator operation is called the Lie algebra of � . The Liealgebra contains essential information about � . Indeed one can reconstruct theconnected component of the identity element in � just from a knowledge of theLie algebra. The lack of commutivity in the group operations corresponds to thenonvanishing of the Lie algebra commutators. For more details on the relationshipbetween Lie groups and Lie algebras see [[43]], [[52]], [[78]], [[79]].)

Recall that � � � � � � 9 � � � � � �� . Corresponding to the representation � � of � �we can define the analogous operators � � where

� � � ����� � � � � � � � � � � ����� � � �� � � � � " � �and

� ( * � � � � is a�

function with compact support. From (4.5) we see that

� 4 � �� � � � � � " � � � � � � � � " � � � (4.14)

Defining the commutator of operators� ��� by � � ��� � � � � � � � we verify

that ��� 4 � � � � �� �� � � " � � � � � � " � � � � � � �

��� 4 � � � � � � �� � � " � � � � �� � (4.15)

�� � � � � � � � " � � � � � " � � � � ��� �63

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where�

is the zero operator, in analogy with (4.13). Applying these operatorson the domain � of

� functions with compact support, a dense subdomain of* � � � � , we see that they are skew-adjoint; i.e., � �� � � � � , � � � " � � , where the

adjoint � � of � is defined by��

� 4 � � � � � � � � � � 4 ����� � � � ����� � �&� � � 4 � � � � � � (4.16)

for all� 4 � � � ( � . (In the case of � 4 we have to integrate by parts.)

The skew-adjoint operators

� 4 � �� � � � � � " � � � � � � � � " � � � (4.17)

satisfy the Heisenberg commutation relations (4.15) and are reminiscent of theannihilation and creation operators for bosons, familiar from quantum theory.In this theory there is a separable Hilbert space

�, an annihilation operator �

and its adjoint the creation operator � � such that

��� � � � � � � � (4.18)

where � is the identity operator on�

. Here � and � � , and the relation (4.18) arewell-defined on some dense subspace � of

�and map � into itself. It is further

assumed that the equation �� � , � ( �

, has a unique solution in � , up to amultiplicative factor. The normalized solution

� �, � � � � � � � � is called the vacuum

state. Finally it is assumed that the closure of the subspace generated by applying� and �

�to

� �recursively, is

�itself. With these assumptions one can construct

explicitly an� �

basis for�

, a basis of eigenvectors of the number of particlesoperator � � � � � .

To make the commutation relations of the � -operators agree with (4.18) andto assure that �

�is the adjoint of � we have, in essence, only one choice:

� � � 4� ���� % � � � � 4� �

�� 4 � ��� � � � �

� � 4� ��� �� % � � � � 4� �

�� � 4 � ��� � � � �� � � ��� � � � (4.19)

To find the vacuum state� � �����

we solve the equation �� ��� . The solution of

this first order differential equation is easily seen to be

� � ����� � � � 4 8 < � � % 8 �64

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where the constant factor is chosen so � � � � � � � � . Since

��

���� � �" � �� � � �

� �" � �" (4.20)

we have �� � � . (We can take � to be the space of all functions � ����� � � % 8 �

where � is a polynomial.) Now let�

be a normalized eigenvector of � witheigenvalue � . The commutation relations (4.18) imply

���� � � � � � � � � � � � �

���� � � � � � � � � � (4.21)

Thus ��

and � are raising and lowering operators: Given an eigenvector witheigenvalue � we can obtain a ladder of eigenvectors with eigenvalues � � � , � aninteger. Now �

�� � � �

� � � � �� �� � � � � � � � � � � � � � � � � � � � � � (4.22)

Thus for � � �, � � � � � � � � � � � ��� �

, so the process of constructing eigenvec-tors of � by applying recursively the creation operator to the vacuum state can becontinued indefinitely. Indeed from (4.21) and (4.22) we can define normalizedeigenvectors

�� with eigenvalues � recursively by

�� �

�� �

� � � � 4 8 � � � � 4 � � � � � � � " � . . . (4.23)

The commutation relations imply the formulas

����

��� � �

���

� 4 8 � � � � 4 (4.24)

Substituting expressions (4.19) into (4.23) and (4.24), we obtain a second-orderdifferential equation and two recurrence formulas for the special functions

�������

.We can obtain a generating function for the

�� from the first-order operator � � �4� �

��� % � � � . Note that � � � 4� � � % 8 � � �� % � � � % 8 � . Hence by Taylor’s theorem,

� 2 ��� � ����� � � � 2 � � � � � � � % 8 � � � � �� � �

��� % � � � � % 8 � � ����� �� � % 8 � � � '2% � �

� , 8 � � ��� � " � 4 8 � 4 �� ��� � � 2 < ��" � 4 8 � 4 ��� � ��� � " � 4 8 � 4 �for any analytic function

�. On the other hand, from (4.23) we have

� 2 ��� � � �����&� �

� �� �

� ���� � 4 8 � 4 � � � � #������

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Comparing these equations, we find the identity

��� �� 0 �" � 0 � � �

���� � 0 �&�

�� " � � � � �

� � � 4 8 � 0 �

�� � #������ (4.25)

In the special case �� �

, the generating function yields

� � 4 8 < ��� ) � 0 � ��"�0 � � �" � � / � � "

8 � 0 � � � 4 8 � � #������ (4.26)

Comparing this with the well-known generating function

��� � � 0 � � "�0 ��� � � 0

� �����

for the Hermite polynomials � #����� , [[40]], [[78]], [[79]], [[80]], [[107]], [[110]],we obtain

� ����� � � � 4 8 < � � � � 4 8 � � ��� � " � 8 � � � % 8 � � #������ (4.27)

The above series converge for all�

and 0 . Since the $ � � ����� ' form an���

set in* � � � � we easily obtain the formula� � � ������ � #����� � � % � �&� � 4 8 � " � � � - �

We sketch a proof of the fact that the $ � � ����� ' form an���

basis for* � � � � . It

is enough to show that this set is dense in* � � � � , i.e., if

� � � � � � � � � � � � � (+* � � � � � �� � � � � " � . . . � (4.28)

then �������� �

almost everywhere. Since � ������

is a polynomial of order � in�

with the coefficient of� �

nonzero, conditions (4.28) are equivalent to� � � � � � � � ����� � �&� � � �� � � � � " � . . .

Now consider the function

� � � ��� � � � � % � � � � ����� � �� Since �

( * � � � � , � � � � is an (entire) analytic function of � and its derivatives canbe obtained by differentiating under the integral sign:

� � � � � �� � � � � ' � , � � �&� � � �� � � % � � � � � � ����� � �� 66

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Now

� � � �&� � � � '

� , � � �� � � � 0 �

Since � � � � is the Fourier transform of � � � � ����� , it follows that ������ � �

almosteverywhere.

4.6 The Bargmann-Segal Hilbert space

Our next task is to compute the matrix elements �� � 9 � � � � � � 9 � � � � � � withrespect to this basis. However, a computation of these matrix elements using thedirect evaluation of the integral is not very enlightening. A better approach is touse a simpler model of the representation � 0 � � , motivated by another solutionof the Heisenberg commutation relations � � � � � � � � � . Rather than the solution(4.19) we can try

�� � � � � � �� � (4.29)

This will work provided we can define a Hilbert space�

on which � and � �

act and such that ��

is the adjoint of � . (We will construct an���

basis $�� � 'for

�corresponding to the basis $ � � ' .) Since ��� � � � ��� �

implies that � � � � � isconstant, in order to mimic successfully the construction (5.22), (5.23) we seethat the � � must be proportional to � � , so the elements of

�must be functions

� � � � . If � were a real variable it would not be possible to find an inner prod-uct

� � 4 � � � � � �� 4 � � � � � � � � � � � � � � with respect to which � � is the adjoint of � .

However, if we take � to be a complex variable and search for an inner prod-uct of the form

� � 4 � � � � � � � � � � 4 � � � � � � � � � � � � �� � � � � � we will be successful.(Here � � � � � � and the region of integration is the plane

� � . We assumethat the weight function

�is nonnegative.) Integrating by parts in the formula� � 4 � � � � � � � � � � 4 � � � � , assuming that the boundary terms vanish and that the for-

mula holds identically in � 4 � � � we obtain the condition � ���� � � � � �� � � � � � � � �� � .Thus� � � � �� � � � � 4 � � ���� where we have chosen the constant � � 4 so that

� � � � � � � .It follows from the analogs of (4.23), (4.24) that

��� � � � ��� � � 4 � �� � � � � � � ��� � � 4 ��� � �

�� � � � � � � � � � � -� �

� � � � � � � � " � . . . (4.30)

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where � � 4 0 and

� � � � �&� � �� ���� �

� � � � � " � . . . (4.31)

For any two functions � � � � � � � � � and � � � ��� � � � � in�

we find

� � ��� �&� � � � � � (4.32)

and � � � � � � � � � � � �&� � ��� � � � (4.33)

From (4.33), � belongs to the Hilbert space�

if and only if� � � � � � � � � .

Clearly if � ( �then there is a constant

� � �such that � � � , � � � for all

. By the ratio test, the series� � � � converges for all complex � , so � is

an entire function, i.e., the power series expansion for � has an infinite radius ofconvergence.

The space�

was introduced by Segal and studied in detail by Bargmann[[13]]. We mention here some of the special properties of

�.

Define the function � � ( �for some complex constant � by � � � � � � ��� � � � � �&�� � � �� � � � . It follows from (4.32) that for any � ( �

,� � � � � � � � � � � � � � � � �� � � � � � � � � � � � �&� � � � � Thus � � ( �

acts like a delta function!From the Schwarz inequality, � � � � � � � � � � � � � � �-, � � � � � � . � � � � � � � � � � 8 � � � � � � . Thus,

if � ��� ( �, then � � � � � � � � � � � , � � � � 8 � � � � � � � � which shows that convergence in

the norm of�

implies pointwise convergence, uniform on any compact set in�

.Now we construct a representation of ��� on

�, using the annihilation and

creation operators (4.29). Comparing with (4.19) we see that the standard basisfor the Lie algebra of � � is

� 4 � 4� � � � � � � ��� 4� ��� � �

�� �

� � � � ��� � � � � � � � � �&� � ��� � � ��� � �

�� �

� � � " � � ��� (4.34)

Mimicking the derivation of (4.25) by exponentiating these operators, we obtain

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the following candidates for operators defining a unitary rep of � � on�

:

� � � � 4 � � � � � � � � � � ��� �� � �< � " � 4 8 � � 4 � � � � � �$" � 4 8 � � 4 � �� � � � � � � � � � � � � � � ��� �� � � � � � �� ��" 4 8 � � � � � � � � � � � ��" 4 8 � � � � � � � �� � � � � � � � � � � � � � � � ��� � ��� � � � �� (4.35)

Since 9 � � � � 9 � � � � � � � � 9 � � 4 � � � � � 9 � � � � � � � � we construct the operators� ��� � � 0 � ��� 9 � � � � by

� � � � � � � � � � � � � � � � � � � � � � � � � 4 � � � � � � � � � � � � � � � � � � � �� ��� � � ' � � � < � � � ,< � ' � � � �� ;� � � ,� � � � � � � � 4 � � � " � � � � � � ���� � ) � � � �� ;� � � *� � �

(4.36)It is straightforward to check that the operators � ��� � � define a unitary rep of � � on�

. As one would expect, this rep is equivalent to the rep � 0 � � of � � on* � � � � .

To establish the equivalence we construct the unitary operator� % * � � � ��� �

which maps the���

basis vector� �����

of* � � � � , (4.27), to the

���basis vector

� #� � �&� � � � of

�:

� � � � � � � � � � 9 � � � ��� � ����� � � � � ( * � � � � �9 � � � ��� � � � � � � � � �����&� � � 4 8 < ��� � � � � � � � � � #" � � " � � � (4.37)

The last identity follows from (4.26) with � � � "�0 . Since 9 � � � . � ( * � � � � foreach � ( �

the integral in (4.37) is always defined. Now if� ( * � � � � then

it can be expanded uniquely in the form� � �

� � � � � � . From (4.37) and theorthogonality of the basis $ � ' for

* � � � � we have

� � � � � � � � � � � ( �

where the $�� ' form an���

basis for�

. Clearly the operator�

is unitary. Acorrect expression for

� � 4 %�� � * � � � � is a bit more involved:

� � 4 � �����&� � � �� � 4 � � � 4

� 9 � � � � ��� � � � � � � � � �� � � � � � �see [[13]], [[78]]. (It is necessary to insert the parameter � � � because 9 � � � ���for fixed

�does not belong to

�.)

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Now we can verify explicitly the relations� ��� � 4 � � � � � � � � � � � 4 � � � � � �� � � � � � � � � � � � � � � � � � � � � � �� ��� � � � � � � � � � � � � � � � � � � � �where � 0 � � is given by (4.5). Since

�is invertible, it follows that � � � � � �

� � � � � � � 4 , so � � is equivalent to � .Since our chosen

���basis for

�consists simply of powers of � , it is relatively

easy to compute the matrix elements � � � � � � � ��� � � � � � � � . (It is immediate that� � ��� � � � � � � �� � � � � � � � � � � � so these matrix elements are exactly the same asthose which could be computed using the

� �basis $ � ' for

* � � � � .) We definethe generating function

� � � � 6 � � � � � � � � � � � �� � � � �&� � � � � � � � � � � � � � � � � 6 � � �� � � ��� (4.38)

Due to the delta function property of � �� we obtain� � ��� � � � �� � � � � � � � � � � � �� � � � �� ��� � � ' � � � < � � � ,< � ' � � � ��� � � ,� � � � � � � � 4 � � � " � � � � � �� ��� � 6 � � � ) � � � ��� � � *� � � �

(4.39)Introducing polar coordinates

� 4 � � � � � " � � � � � � ��� and equating coefficients of

6 � � � in (4.38) and (4.39) we obtain the explicit ex-pression

� � � � � ��� � " � � � � � � � � � � � � � � � � 4 � � � � � � 8 < � � � 4 8 � �� �� ��� � � � * ' � � , �� � #" � (4.40)

where* ' 2 , � � � is the associated Laguerre polynomial [[40]], [[77]], [[78]]. An

alternate expression is

� � � � � ��� � " � � � � � � � � � � � � � � � � 4 � � � � � � 8 < � � � 4 8 � ��� �� ��� � � � * ' � � ,� � � #" �� (4.41)

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Note that the skew adjoint operator � � � � � � � ��� is well defined on

�and

can be exponentiated to yield the unitary operator �� � 4 � :

�� � 4 � � � � ��� ��� �

� �'4 � �� � � � � � ��� � � � � 32 � � � � ( � (4.42)

Here the eigenvectors of � acting on�

are just the� �

basis vectors � � :

�� � 4 � � � � � � � 2 � � (4.43)

(One can extend the rep � � of the three parameter group ��� to the four-parameteroscillator group generated by � � � � � and �

� 4 � . See [[77]], [[78]] for the details.)Using the unitary transformation

�one can transform the �

��� 4 � to unitary opera-tors on

* � � � � :�� 4 � � ����� � � � � � � � �� � � � ��� � � � ��� � ��';��� � � ��� 2 � , ��� ���� � � � ��� 4 ��� � �� � � #" � � � � � ��� 4 � � � � � � � (4.44)

Here, �� 4 ��� � � 4 � � � 4 � � and 4 � " � �� 0 ,

an integer,

� � � , � � 0 � � .(See [[13]] for details.) Note that �

� � " � is just the (ordinary) Fourier transformon

* � � � � . Thus, the Fourier transform is embedded in a one parameter groupof transformations. the infinitesimal generator of this one-parameter group is thesecond-order differential operator

� � � � �� �" � �� � � �

� �" � �" � Furthermore, �

� 4 � � � � � � � 2 � � .It follows from the orthogonality relations (4.10) that the matrix elements

(4.40) form an� �

set in* � � � � � . (See [[77]] or [[78]] for a direct proof of

this fact.) These orthogonality relations reduce to the following orthogonality re-lations for associated Laguerre polynomials:� � * ' ,� � � �%* ' ,� �� � � � � � � � 4 � � �

� � � �" ��� -

� � � (4.45)

valid for all integers � � � � �and all integers

such that � �

� � � � � ��. (By switching between (4.40) and (4.41) depending on whether

� � , �or

� � � �we can always take

� �in (4.45).) Since for each

� �the associated Laguerre polynomials are known to be complete in

* � � � � � � with

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weight function � � � � � 4 , it follows that the matrix elements $� � � ' form an���

basis for* � � � � � , with the usual Lebesgue weight function � . (Indeed, this follows

from Exercise 5.2, without any knowledge of the completeness properties of theLaguerre polynomials.) Thus, the matrix elements of � with respect to any

� �basis for

* � � � � will form an���

basis for* � � � � � . (In other words, the cross-

ambiguity functions with respect to an���

basis of signals $ � ' form an���

basis for* � � � � � .)

4.7 The lattice representation of� �

There is another realization of the irred unitary rep � � that we shall find useful:an induced rep of ��� from the subgroup �

�where

�� � ��� �� 9 � � 4 � � � � � � �&� ��� � � 4 � �� � � �� � �

� � � ���� �� 4 � � � are integers and � � ( �

. Note that the operators � � � � 4 � � � � � � ��� � ���� � �define a one-dimensional unitary rep of �

�. (Here, the fact that � 4 � � � are integers

is crucial in verifying that � �is a rep of �

�.) We will study the rep

)� of � �induced from the rep � �

of ��, (4.2)-(4.4). Here

)� is defined on the space � offunctions � on � � such that � � � 9 �&� � � � � � � � 9 � for all � ( � � � 9 ( � � , i.e.,

� � � 4 � � 4 � � � � � � � � � � � � � � 4 � � �&� � ���& � � � � � 4 � � � � � � �� (4.46)

The operators)� � 9 � � 9 ( � � act on � according to

� )� � 9 � � � � 9 � � � � � 9 � 9 �� (4.47)

We see from (4.46) that for any 9 � � 4 � � � � � � � we can always choose � � � 4 � � � � � � �such that � 9 � 9 � � � � 4 � � � � � � � where

� , � � 4 � � � � , � � � � � . Thus � can berestricted to � � � � � � � with coordinates

� � � 4 � � � � � � � . Moreover, setting � � �� � � � � � � 4 � � in (4.46) we have the periodicity condition

� � � 4 � � 4 � � � � � � � � � � ���& � � � � � � 4 � � � � (4.48)

where � � � 4 � � � � � � � � 4 � � � � � � . Conversely, given � satisfying (4.48) we candefine a unique � satisfying (4.46) by

� � � 4 � � � � � � ��� � � � 4 � � � � � ���� ��� 72

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The � � -invariant inner product on � is � � 4 � � � :� � 4 � � � � � � 4� � 4� � 4 � � 4 � � � � � � � � 4 � � � � � � 4 � � � � (4.49)

and the operator)� � � � 0 )� � 9 � � 4 � � � � � � � � acts on these functions by� )� � � � � � � � 4 � � � �&� ��� � " � � � � � � � 4 � � � � � � � 4 � � 4 � � � � � � �� (4.50)

To recapitulate, we have defined a unitary rep)� of � � on the Hilbert space� � of all functions � satisfying (4.48) and of finite norm with respect to the inner

product (4.49). This is known as the lattice representation of � � .The lattice rep is equivalent to the irred Schrodinger rep � 4 , (4.5). To see this

consider the periodizing operator (Weil-Brezin-Zak isomorphism)

� � � � 4 � � � � � � � � � � � � � 4 � � 4 � � � � � � � � �'� � �� � ���� ��� �

� � � ���� � � � � � � � 4 � (4.51)

which is well defined for any� ( * � � � � which belongs to the Schwartz space. It

is straightforward to verify that � � � �satisfies the periodicity condition (4.46),

hence � belongs to � . Now� � � � . � . � � � � � � � � . � . � � � �� � 4� � � 4 � 4� � � � � �&� � � � ���& (' � � � , � � � � � � 4 � � � � � � � 4 �� � 4� � � 4 �

� � � �� � � 4 � � � � � � � 4 �&� � � � ��� 4 � � � ����� � �� � � � � � �

so � can be extended to an inner product preserving mapping of* � � � � into � .

It is clear from (4.51) that if � � � 4 � � � � � � � � � 4 � � � � � � then we can recover� � � 4 � by integrating with respect to � � % � � � 4 � � � 4� � � � 4 � � � � � . Thus we definethe mapping � � of � � into

* � � � � by

� � � �����&� � 4� � ��� � � � � � � � ( � � (4.52)

Since � ( � � we have

� � � ��� � � �&� � 4� � ��� � � � � � ���� �9� �=� � �� � � �����for � an integer. (Here �� � ����� is the � th Fourier coefficient of � ��� � � � .) The Parsevalformula then yields � 4� � � ��� � � � � � �=� �

� � � � � � ��� � � � � �73

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so � � � � � � � 4� � 4� � � ��� � � � � � � � � � � � 4� � � � � � � � ��� � � � � � � �� � � � � � � ����� � � � � � � � � � � � � � ��

and � � is an inner product preserving mapping of � �into

* � � � � . Moreover, it iseasy to verify that � � � � � � � � � � � � � �for

� ( * � � � � , � ( � � , i.e., � � is the adjoint of � . Since � � � � � on* � � � � it

follows that � is a unitary operator mapping* � � � � onto � � and � � � �

� 4 is aunitary operator mapping � � onto

* � � � � .Finally,� � � 4 � � � � � � � � � � ���& (' ��� � � � � � � � , �

� � � ���� � ' � � � , � � � � � 4 � � 4 �� � )� � � � � � � � � �so � � 4 � � � � )� � � � � and the unitary reps � 4 and

)� are equivalent.

4.8 Functions of positive type

Let � be a group. A complex function�

on � is said to be of positive typeprovided for every finite set � 4 � . . . � � # of elements of � and every set of complexnumbers

� 4 � . . . � � # the inequality � � � � � � � 4� � �

� �� � � � � � (4.53)

holds. If � is a unitary rep of � on the Hilbert space�

then the inner product� ��� � �

���� � � �/� is of positive type on � for every

� ( �. Indeed,� � � � � � � � 4� � �

� �� � � � � � � � � � � � � � 4� � �� � � �/� �� � � �� � � � � � � � � � � � � � � � � � �/� �� � � �� � � � � � � � � � � � � � � � � � � � � � � � � � (4.54)

It follows from this construction, in the case where � � � � , that narrow bandambiguity functions are of positive type on � � .

It is an important result of abstract harmonic analysis that all functions ofpositive type on a group arise as diagonal matrix elements of unitary reps, exactlyas in the construction (4.54). This result, whose proof we now sketch, sheds lighton the structure of the set of ambiguity functions.

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Note first that the inequality (4.53) implies that the � � �

matrix with ele-ments � � � � � � � � 4� � �

�is Hermitian

�� � � � �

� � � � and nonnegative. Thus the-�

eigenvalues of this matrix are nonnegative; hence the determinant is also nonneg-ative.

Lemma 2 Let�

be a function of positive type on the group � . Then1)� ��� 4 � � � � � �

2)� � � � � � � � � � �

for each �( � .

PROOF: For �!� � , the inequality (4.53) yields

� � � � � �. Now take

��� " ,� 4 � � , � � � � . Then � � � � �&�

�� � � � � �

��

� ��� 4 � � � � � �

Since � 4�� � �� �74 we have

� ��� � � �

�� 4 � . Further, � � � � � � � � � � � � � � � � � � 4 � ��

. Q.E.D.

Theorem 14 Let�

be a function of positive type on � . Then there is a unitaryrepresentation � of � on a Hilbert space

�such that

� ����� �

���� � � �/� for some

� ( �. Furthermore the span of the set $�� � � � � % � ( � ' is dense in

�.

PROOF: We will use the computation (4.54) as a motivation for the constructionof�

and � . Let*

be the subspace of the group ring� �

consisting of thoseelements � which take on only a finite number of nonzero values:

� �

� . �

(Alternatively we can consider � as a function �% � � �

which is zero except ata finite number of points � .) The inner product of two vectors in

*is defined as� � � � � �

� �� ��� 4 � � � �� � �

where we sum over all points � � such that either ��� ��

or � � � � � is nonzero. It isevident that

� . � .3� is linear in its first argument and from Lemma 2 that� � � � � �� ��� ��� and

� ��� � � � � .Thus,

� . � .3� satisfies all requirements for an inner product, except that we mighthave

� ��� � � � �with � �� . It follows from this result that the Cauchy-Schwarz

inequality is valid: � � ��� � � � � , � ��� � � � � � ��� .75

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We use a standard construction to convert� . � .3� to a true inner product.

Let� � $ � ( * % � ��� � � � � '

. Then�

is a subspace of*

. From theCauchy-Schwarz inequality we have

� ��� � � � �for �

( �, � ( *

. Thus� � 4 � � 4 � � � � � � � � � � 4 � � � � (4.55)

for � � ( * � � � ( �. We can now define

� . � .3� on the factor space* � whose

elements are the sets �� � � � � $$� � � % � ( � '

. The set � � �corresponds to the zero vector. From (4.55) we see that the definition�

� 4 � � � � � � � 4 � � � � � � � � � � � 4 � � � �is unambiguous. Furthermore

�� � ��� � �

only if �� � � , i.e., �

� . Thus� . � .3� is a true inner product on* � . Now by taking all Cauchy sequences in

* �we can complete

* � to a Hilbert space�

.Given � ( � � � � � � .�� ( *

we define the linear operator �� � � % * � *

by �� � � � � � � . � � . Then�

�� � � � � � � � � � � � � � � � � � � 4 � � 4 � � � � �� � �� � � � � � � � 4 � � � �� � � � � � � � �

so �� � � is an isometry on

*. Furthermore, �

� � � 4 � � � � � � ��� � � for all �(�*

, so�� � � is invertible, hence unitary on

*. Also, �

� � 4 � � � � � �� � 4 � � � � � � � ��� so � is

a unitary rep of � on*

. Clearly, the operators �� � � extend uniquely to a unitary

rep of � on�

.Let

� � � . � ( *. Then the vector $�� � � � � ' span

*, for if �

� � � . � we have �

� � � � � � � � . Moreover,����� � � � � � � � � � . In the extension of* � to

� � � maps to � ( �such that the span of $�� � � � � ' is dense in

�and�

���� � � �/� � � � � � . Q.E.D.

If � is a linear Lie group, one can show that the function�

is continuous on �if and only if � is a continuous rep of � , [[43]], [[83]].

The unitary rep � constructed in Theorem 14 is unique up to equivalence.Indeed, suppose there are unitary reps � 4 � � � on Hilbert spaces

� 4 � � � such that�� 4 � � � � 4 � � 4 � 4 � �

� � � � � � � � � � � � where the spans of $�� � � � � � � ' are dense in� � .

Define the map� % � 4 � � � by

� � � � 4 � � � � 4 �&�

� � � � � � � � (4.56)

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for all finite sums � �4 � � � � 4 � � � � 4 . In particular� � 4 � � � . This mapping is

well defined because if � �4 � and� � �4 � � �� then�

� �� � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � 4� � � � � � � � � � � �� �� � � � � � 4 � � � 4� � � � 4 � � 4 � 4 � �� � � �� �4 � � �4 � 4� � �

so � �� � . (Furthermore this same calculation shows that�

is an isometry.) Thus,�extends to a unitary transformation from

� 4 onto� � . Finally

�� 4 � � � � �4 �

� � � � � � �� � � � � � � � � �4 for all finite sums � �4 , so�� 4 � � � � � � � � ��� , and the reps � 4

and � � are equivalent.

Corollary 4 Let � be a unitary irred rep of the group � on the Hilbert space�

such that ����� � 4 � � 4 � � �

���� � � � � � �

for all �( � , where � 4 � � � are nonzero elements of

�. Then � � � � � 4 for some� ( �

with � � � � � .PROOF: Since � is irred, the spans of $�� � � � � � ' are dense in

�. From the con-

struction (4.56) with� 4 0 � � and � 4 � � � 0 � � � � � we see that there is a unitary

operator� % � � �

such that����� �

������

for all �( � and

� � 4 � � � .Since � is irred it follows from Theorem 12 that

� � � � , where � � � � � since�

is unitary. Thus,� � 4 � � � . Q.E.D.

Note that the corollary implies that two signals correspond to the sameambiguity function if and only if they differ by a constant factor of absolutevalue one.

Next we will characterize those functions of positive type on a topologicalgroup � that correspond to irred unitary reps of � . Consider functions

� 4 � � � ofpositive type on � . We say that

� 4 dominates� � if� 4 � � � is of positive type on �

(so that� 4 � � � � � � 4 � � � � is a sum of two functions of positive type). A function�

of positive type on � is indecomposable if the only functions of positive typedominated by

�are scalar multiples of

�. (Clearly, the multiples must be of the

form � � where�� � � � . )

Theorem 15 The function�

of positive type on the topological group � is inde-composable if and only if � �

�� � �

���� � � � � (4.57)

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for all �( � where � is a unitary irred rep of � on some Hilbert space

�and

� ( �.

PROOF: Suppose�

is indecomposable. By Theorem 14 there is a unitary rep �

of � on the Hilbert space�

and a vector � ( �such that the span of $�� � � � � '

is dense in�

, and� ��� � �

���� � � �/� . Let

�be a nonzero subspace of

�which

is invariant under �%���� � � �

for all �( � . Then

� �is also invariant

under � (see Theorem 9), and we have the unique decomposition � � � 4 � � �with � 4 ( � � � � ( � �

. It follows easily that� ��� � �

���� � � �/� � �

���� � 4 � � 4 � ��

���� � � � � � � � � 4 � � � � � � � � � . Since $�� � � � � 4 ' is dense in

�,� 4 � � � � �

���� � 4 � � 4 � is

a function of positive type and�

dominates� 4 . Hence

� 4 � � � for some constant� �� �

, so����� � 4 � � 4 � � �

���� � � � 4 � � �

���� � � � �/� or,�

���� � � � 4 � � �/� � �

(4.58)

for all �( � . It follows that � 4 � � � so � ( �

, hence� � �

and � is irred.Conversely, suppose there is a unitary irred rep � of � on

�such that (4.57)

holds for some � ( �. Let� 4 be a function of positive type on � that is dominated

by�. Without loss of generality we can assume that � is obtained from

�and

the space* � according to the construction given in the proof of Theorem 14.

On this same space we can construct the inner product� . � .3� 4 associated with the

function� 4 . Since

� 4 is dominated by�, the Cauchy-Schwarz inequality implies� � ��� � � 4 � � , � ��� � � 4 � � � ��� 4 , � � � � � � � � �0� � � for all ��� � ( * � . Thus

� . � .3� 4 extendsto a positive definite Hermitian form on

�such that� � � 4 � � � � 4 � � , � � � 4 � � � 4 � � � � � � � , � � � 4 � � � � � � � � � � (4.59)

This means that there exists a bounded self-adjoint operator�

on�

such that� � 4 � � � � 4 � � � 4 � � � � � (4.60)

for all � 4 � � � ( �. (Indeed, it follows from (4.59) that for fixed � � � � � 4 � � � � 4 is

a bounded linear functional�

. By the Riesz representation theorem [[95]] thereexists a vector ���

( �for all � ( �

such that� � 4 � � � 4 � � � 4 � ��� � . Clearly, the

map � ���� is linear, so there exists a linear operator

� % � � �such that

���� � � . Since � � � 4 � � � � � � � � � � � 4 � � � � 4 � � , � � � 4 � � � � � � � � � � , we have in the case

� 4 � � � � the inequality � � � � � � � < , � � � � � � � � � � � � � � � or � � � � � � � � ,�� � � � � � � . Thus,�is a bounded operator. Furthermore,

� � 4 � � � � � � � � 4 � � � � 4 � � � � � � 4 � 4 �� � � � � � 4 � � � � � 4 � � � � , so�

is self-adjoint. Since� � � � � � � � � � � � 4 � � � � is

nonnegative.)

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By construction of� . � .3� 4 through the completion of

* � we have����� � 4 � � � � � � � � 4 �� � 4 � � � � for all �

( � . Hence,����� � 4 � � �

��� � � � � � � 4 � � � � � and �

��� � 4 � �

��� �

�, which implies

�����&�

���� �

for all �( � . Since � is irreducible, we have� � � � for some positive constant � . Thus� 4 � � � � �

���� � � �/� 4 � �

���� � � � �/� �

� � � � � � � � �/� � � � � � � , so�

is indecomposable. Q.E.D.

Corollary 5 Let � be an irred unitary rep of � on�

, and suppose � � � � 4 � � � arenonzero elements of

�such that�

���� � � � � � � � �

���� � 4 � � 4 � � �

���� � � � � � � (4.61)

for all �( � . Then � 4 � � � � , � ( �

.

PROOF: The functions of positive type� � � � �%� �

���� � � � � � � satisfy

� � ���%�

� 4 � � � � � � � � � , so�-�

dominates� 4 and

� � . Since � is irred,�

is indecompos-able. Thus

�-� � � � 4 � 4 � � �� � � where � 4 � � � are positive constants. Setting � �� � � � � � ,� � � � " we have �

���� � � � � � � � �

���� � �4 � �4 � � �

���� � �� � � �� �

By Corollary 4, � �4 � � � �� . Hence � 4 � � � � for� � � � � � 4 . Q.E.D.

Note that Corollary 5 implies that the sum of two ambiguity functionscorresponding to nonzero signals � 4 � � � is again an ambiguity function if andonly if � 4 � � � � .4.9 Exercises

1. Verify explicitly equation (4.10).

2. Show that if $ � � � � � � � � � " � . . . ' is an� �

basis for* � � � � then the matrix

elements $ � � � � � 4 � � � � � � � � � � ' form an���

basis for* � � � � � . Hint:

From exercise 5.1, the matrix elements form an� �

set. Hence to showthat they form a basis it is enough to prove that if� �� � � 4 � � � ��� � � � 4 � � � � � � � � � � �

� � 4 � � � ��� �for �

( * � � � � � and all � , � , then �� �

almost everywhere. This showsthat the

� �set is also dense in

* � � � � � , hence is a basis.

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3. Set � � � � � � � � � � � � 4 � � � � � � � � for� � � ( * � � � � and � � � � � � � � � � � .

Show that

� � � � �� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� 4. Prove that the ambiguity functions � � � � � 4 � � � � � � � � for all

� ( * � � � �span a dense subspace of

* � � � � � .5. Suppose

� ( * � � � � such that��� �� �

almost everywhere. Prove that theset $ � ���� 5' � � � � � � � , ��� � ��� � � � 4 � � � � � � � � " � . . . ' is an

���basis for the

lattice Hilbert space � � . Find an explicit expression for the corresponding���basis of

* � � � � obtained from the mapping �� 4 .

6. Show that the rep � � of � � is continuous in the sense that

� � � � � � 4 � � � � � � � � � � � � � � � � � � � 4 � � � � � � � � � � � � � � �for each

� ( * � � � � .7. Show that the ambiguity and cross-ambiguity functions � � � � � 4 � � � � � � � �

are continuous functions of� � 4 � � � � .

8. Construct the unitary rep)� � of � � induced by the one-dimensional rep)� � � � � 4 � � � � � � �&� � ���� � � � of the subgroup � 4 , where � is an integer (not nec-

essarily positive). Determine the action of � � on the rep space, in analogywith (4.50). Under what conditions on � is

)� � irred? Show that the repspace contains � ��� linearly independent ground state wave functions.

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Chapter 5

REPRESENTATIONS OF THEAFFINE GROUP

5.1 Induced irreducible representations of� 9

Recall that the affine group � � is the matrix group with elements

9 � ��� � ��� �� �� � � � � � � (5.1)

and multiplication rule � ��� � �'� � � � � � � � � � � � � � � � � � �� Even though this is a " �$" matrix group with a very simple structure, there aresome difficulties in developing its rep theory in accordance with the general resultspresented in Chapters 1-3. An indication of the complications appeared already inChapter 3 where we showed that � � is not unimodular, i.e., � � 9 �� � �%9 . Indeed

� � 9 � � � � �� � � �#��9 � � � � �

(5.2)

We begin by deriving the irred unitary reps of ��� . One family of such repsis evident:

�� � ��� �+� � � � � � real. We use the method of induced reps to derive

several forms of the remaining irred unitary reps.Consider the subgroup � 4 � � �

of elements of the form 9 � � � � � , � (��. The

unitary irred reps of � 4 take the form � � � � � � � � � . We now construct the unitary

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rep of � � induced by the rep � � of � 4 . The rep is defined on a space of functions� � ��� � � on � � such that

� � � 9 �&� � � � � � � � 9 � � � (� 4 � 9 ( � � �

i.e.,� � ��� � � � � �&� � � � � � ��� � � (5.3)

Thus � � ��� � � � � � � � � ��� � � � � � � � � � � where � is defined on the coset space� 4 � � � 4 � � � � � � � and � � is the subgroup of elements 9 � ��� � � � � � �. The

action of � � in the induced rep is given by

� � � � 9 � � � � 9 � � � � � 9 � 9 � � 9 � 9 � ( � �or, restricted to the functions � :� � � � ��� �+� � �'� � ��� � � � � � � ��� � (5.4)

where 9 � � 9 ��� ��� � � . The right-invariant measure on � 4 is � � � � �&� � � � , so theinner product

� . � .3� 4 with respect to which the operators� � � � � � � are unitary, is� � 4 � � � � 4 � � � � 4 � � � �� � � � � � �� (5.5)

The elements of the Hilbert space� 4 with the inner product are Lebesgue mea-

surable functions � � � � such that � � � � � � 4 � � � � � � 4 � � . The rep� �

is reducible.However, we have

Theorem 16 For� �� �

the representation� �

of � � is irreducible.

PROOF: This demonstration is very similar to that of Theorem 13. For complete-ness we repeat the basic ideas of the proof, omitting some of the technical details.Our aim will be to show that if

�is a bounded operator on

� 4 which commuteswith the operators

� � � 9 � for all 9 ( � � then�%� � � for some constant

�, where� is the identity operator. (It follows from this that

� �is irred, for if � were a

proper closed subspace of� 4 , invariant under

� �, then the self-adjoint projection

operator � on � would commute with the operators� � � 9 � . This is impossible

since � could not be a scalar multiple of � .)Suppose the bounded operator

�satisfies

� � � � ��� �+� � � � � ��� �+� �82

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for all real ��� � with � � �. First consider the case �

� � : � commutes with theoperation of multiplication by the function � � � � . Clearly

�must also commute

with multiplication by finite sums of the form��� � � � � � � � and, by using the well-

known fact that trigonometric polynomials are dense in the space of measurablefunctions,

�must commute with multiplication by any bounded function

� � � � on� � � � � . Now let�

be a bounded closed internal in� � � � � and let

� �+( � 4 be thecharacteristic function of

�:

���&� � �&� � � ��� � ( �� � � � ( � Let

� � ( � 4 be the function� � � � ���

. Since� � � � ���

we have� �&� � � �

� ���&� � � � � � � � � � � � ���&� � �%* ���&� � � � ���&� � � � �&� � � so� �

is nonzero only for� ( �

. Furthermore, if� �

is a closed interval with� � � � and

� � � � ��� then� � � � � � � ��� � � � � ��� ��� � � � � ���&� � ��� ��� � � � � � � � � so

� � � � � � � �&� � � for� ( � �

and� � � � � � �

for � ( � � . It follows that there is a unique function� � � �

such that���� � ( � 4 and

���� � � � � � � � � * ���� � � � for any closed bounded interval)�in� � � � � . Now let � be a

� function which is zero in the exterior of

)�.

Then� � � � � � � � � ���� � � ��� � � � � ��� ���� � � � � � � � � � � � � ���� � � � � � � � � � � � � , so

�acts on � by multiplication by the function

� � � � . Since as)�

runs over all finitesubintervals of

� � � � � the functions � are dense in� 4 , it follows that

�%� � � � � � .Now we use the hypothesis that

� � � � � � � � � � � �&� � � � ��� � � � � � � � for all � � �and � ( � 4 % � � � � � � ��� ��� � � ��� � � � ��� � . Thus

� � � �&� � � ��� � almost everywhere,which implies that

� � � � is a constant. Q.E.D.

Lemma 3 For � � � the reps� �

and��� �

are equivalent.

PROOF: Let� � % � � � � �

be the linear unitary operator� � � � � �&� � � � � � . Then� � 4� ��� �

� � and��� � � ��� �+� � � ��� � � � � ��� �+� . Q.E.D.

It follows that there are just two distinct irred reps in the family� �

. Wechoose these reps in the normalized form

� � � � . Thus we have constructed thefollowing irred unitary reps of � � %*� � , � � � � � � � � ; � � and

� � where�� � ��� �+� � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � (5.6)

and � ( � 4 . It can be shown [[67]] that these are the only unitary irred reps of� � .

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5.2 The wideband cross-ambiguity functions

Another important class of unitary reps of ��� can be induced from the subgroup� � � � � �

of elements of the form 9 � ��� � � � � � �. Consider the irred reps � of

� � : � � � � � � , � complex. We construct the rep� � of � � induced by � . It is

defined on a space of functions � � ��� � � on ��� such that

� � � 9 ��� � � � � � � 9 � � � (� � � 9 ( � � �

i.e.,� � � � ��� � � � �&� � � � � � � ��� � �� (5.7)

Thus � � ��� � ��� � � � � � � � ��� � � � � � � where � is defined on the coset space� � � � � � � � � � � � 4 . The action of � � in the induced rep is given by

� � � � 9 � � � � 9 � � � � � 9 � 9 � � 9 � 9 � ( � � �or, restricted to the functions � :

� � � � ��� �+� � �'�����&� � � � � � �� � (5.8)

There is no right-invariant measure on � � . However, � � ����� � � � goes to a multipleof itself, so an inner product

� . � .3� � with respect to which the operators� � � ��� �+� are

unitary is � � 4 � � � � � � � � � 4 ����� � � ����� � � � (5.9)

provided �� � � � 4� where � is real. The elements of the Hilbert space

� � �* � � � � with this inner product are Lebesgue measurable functions � ����� such that� � � � � �� � � � � � � � � � . Since we will be restricting to the case of unitary reps, weintroduce the notation

�� 0 �� �

� � :� �

� � ��� �+� � � �����&� � �� � � � � � �

� � � � ( * � � � �� (5.10)

The rep�

� is reducible. Indeed let us consider the Fourier transform�

as aunitary map from

* � � � � to* � � � � :

�� � � � � �� " �� � � ����� � � ;% � � � � � � � � � (5.11)

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Then� �����&� � � 4 �� �����&� �� " �

� � �� � � � � ;% � � �and the action �� � of � � on the function �� corresponding to the action

�� on � is

�� � � ��� �+� �� � � � � � � �� � ��� �+� � �'� � �� 4� ��� �� � �

� � � � % � �� � � � ;% � � �� � �� 4 8 � � � � �� � � � �� (5.12)

Now consider the map � �%-* � � � � � � �

4 �� �4 defined by

� � �� � � � � � �0� �� 4 8 � �� � � � � � �� �

and let �� � � � � � � � �� � � �)( � �4 � � � ��� ��� � � � � � � � �� � � �)( � �4 � � � � � � � � �

Then the action induced on �� � ( � �4 by

�� � ��� �+� is

� � � ��� �+� �� � � � � � � � � �� � � � � �and the action induced on �� � ( � �

4 is� � � ��� �+� �� � � � �&� � � � � �� � � � � ��

(Here, the spaces� �4 � �

�4 consist of functions

� � � � � � � � � � � on the positive realline with weight functions � � � , � � � , respectively. The operator � � is responsi-ble for the change in weight function.) As the reader can easily verify, we havethe “Plancherel formula”� �

� � ��� �+� � � � � � � � � � � ��� �+� � � � � � � 4 � � � � � ��� �+� � � � � � � 4 (5.13)

where � � � � � � � �� � � � � � � � � �

� � � � � � � �� � � � � � � � � � � � (5.14)

Thus the rep�

� decomposes as the direct sum of the irred reps� � and

� � ;the Fourier component �� � � � of � corresponds to

� � for positive � and to� �

for negative � . (If, however, we restrict the rep� �

to those � ( * � � � � whoseFourier transform �� � � � has support on the positive reals, then

�� is irreducible

and equivalent to� � .)

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Note that the matrix element� � � � ��� �+� � � � ���)� � coincides with the wideband

cross-ambiguity function (1.15) with � � �� 4 � � � � � . Formula (5.13) suggests

that, for computational simplicity, in selecting a basis for* � � � � in which to deter-

mine the cross-ambiguity function, one should choose the union of a basis on thesubspace transforming according to

� � and a basis on the subspace transformingaccording to

� � .

5.3 Decomposition of the regular representation

Even though � � has a very simple structure, the decomposition of the regularrep of � � and expansion formulas for functions on ��� in terms of the matrixelements of irred reps are not trivial consequences of the general theory workedout in the earlier chapters. In particular, ��� is not unimodular, i.e., � �89 �� � � 9 .(The machinery developed in Chapter 3 for averaging over a group assumed thatthe group is unimodular.) Choosing the measure � �89 to be definite and assumingthat the functions � � � � � ( � �

4 are�

with compact support in� � � � � to avoid

convergence problems, we can verify directly that� � � � � � � ��

� � � � � � � � � 4 � � � � 4 � � � � ��� �+� � � � � < � 4 � � � (5.15)

in accordance with the earlier theory, but� � � � � � � ��

� ��� � ��� �+� � 4 � � � � 4 � ��� � ��� �+� � � � � < � 4� " � � � 4 � � � � 4 � � < � � � � � 4 (5.16)

where � � � ������� � � ����� � . To investigate the problem in more detail we will decom-pose explicitly the right regular rep of ��� into irred reps of � � , [V].

Recall that the right regular rep of � � is defined on the Hilbert space� � of

measurable functions� � 9 � ��� � � � 0 � � ��� � � , square integrable with respect to the

measure �#��9 � � � � � � . The inner product is� � 4 � � � � � � � � � � � � ��

� 4 � ��� � � � � � ��� � � � (5.17)

and � � acts on this space in terms of the unitary operators� � 9 � �

:� � 9 � � � � 9 � � � � 9;9 � ��

(5.18)

To decompose� � into irreducible components we project out subspaces of func-

tions which transform irreducibly under the left action of the subgroup � � �86

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$ � � � � � 9 � � � � � ' . (Since the left action of � � commutes with the right action,it follows that these subspaces will be invariant under the operators

� � 9 � �.) The

function�

�� � � � � �,� � �

� 4 8 � � � (%�, defines a one-dimensional rep of � � . Now

consider the map� � � �

where� ( �

� , given by

� � � 9 �&� �� � � � 9 � � �

� � � � � (5.19)

where � � � � � � � � � is the two-sided invariant measure on � � . Note that� �

satisfies� � � � � 9 � � � � � � �

� 4 8 � � � � 9 � . In terms of coordinates we have

� � � � � � � � � � � � � ��� � � � � � �� 4 8 � � � �

� � � � ��� � � � � � �� 4 8 � � � � ��� � �� (5.20)

Note also that the map� � � �

is invertible:

� � � ��� � � ��� �" � �� � � � ��� � � � �� 4 8 ��� �

(This is just the inverse formula for the Mellin transform, a variant of the Fouriertransform, [[41]], [[107]].)

Now (5.20) agrees with (5.7) with �� � � �/� #" , so the action of � � on

the functions� �

induced by the operator� � 9 � is just

� � �� 4 8 � , or in terms of the

functions � ����� where� � � ��� � �&� � �

� 4 8 � � �

��

� � � (5.21)

it reads � �� � ��� �+� � � � �����&� � �

� 4 8 � � �

� � � �� � � (5.22)

in agreement with (5.10). Furthermore, we have the decomposition formula� � � � � � � ��

� 4 � ��� � � � � � ��� � ��� �" � � � � � � � � � 4 ����� � � � ����� � � (5.23)

where the � �� ( * � � � � are related to� � ( � � via (5.20) and (5.21). On each

space of square integrable functions � � � � � � � � � , the rep�

� decomposesinto the direct sum of the irred reps

� � and� � , (5.12) and (5.13). Thus the right

regular rep�

decomposes into a direct integral (rather than a direct sum) of acontinuous number of copies of the irred reps

� � and� � . (The one-dimensional

unitary reps�� � ��� �+� � � � do not appear in the decomposition of

�.)

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As we have shown� � ��� � �&� 4'2��� , � � � � � � � � � �*� � � � � � � �

� 4 8 � �$ � �� � � � � � � � � � � � � � � � � � � � � � � ' � � (5.24)

where � � � � � � � ��� � � �� � � � � � We can express these results in another form by choosing an explicit

� �basis$

?��� � � ' for

4 in the reps� �

. Then

� �

� � � � � � � � �

?��� � � (5.25)

where� �

�� � � � �

� � � �?��� � � � �� (5.26)

Substituting (5.25) and (5.26) into (5.24), using (5.20) and (5.14) to express theexpansion in terms of $

?��'

and�

alone, and making appropriate interchanges ofintegration and summation orders, we arrive at the formula

� � � � � � � 4��� � � � � � � � � � � ��� ? �

������ � � � � � � � �

? ������� � 4� � � � � � � ��� ? �

������ � � � � � � � �

? ������� � 4 �� 4��� � � ��� � � �� � ��� �+� �

� � � ��� ���� (5.27)

Here � ���?� � �����&�� ?

������

, � ? � ;� 4 � � � ? ����� � ����� � ��and

�� � � �

is the operator

�� � � � � �

� � ���� � 9 � ��� � 9 � � � 9 � � � 9 � � � � �

� � (5.28)

(Since the bases $?�

�'

are the same for all � , dependence on this parameter dis-appears from the final result.)

Similarly, the Parseval formula� � � � � � � �� � � � ��� � � � � � �" � � � � �

� � � � � ��� � � � � � �

�� � � �

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yields� � � � � � � �� � � � � � � � � � � �" � � ��� � � �

� � - � ��� � � � � � � � � - � ��� � � � � (5.29)

where- � � ��� � � � � 4 8 � � � ��� � � . (The simple derivation of (5.27) and (5.29) given

here is motivated by Vilenkin’s treatment of the affine group, [[107]]. It is notour purpose here to give a rigorous derivation with precise convergence criteria.Rather, we want to demonstrate simply how group theory concepts lead to thecorrect expansion formulas. A rigorous, but much more complicated, derivationwas given by Khalil [[66]], and the relevance of these expansions to the radarambiguity function was pointed out by Naparst [[84]].)

The expansion (5.27) has been expressed in terms of the inner product onthe Hilbert space

� 4 and the operators� �

. It is enlightening to re-express itin terms of the operators

� �, (5.10) and the Hilbert space

* � � � � . For this we set$?�

������&� � �

? ������� '

where $? ��'

and $? ��'

are each� �

bases for* � � � � . Now

let � �� � � ��� � � 4? ��� � �&� �� " �

� � � ;% �? ������� � �

and � �� � � �&� � � 4? ��� � �&� �� " �

� �� � ;%

� ? ��� � ��� � � �

i.e., � �� are the inverse Fourier transforms of

? �� . (Recall from the discussion

following (5.12) that

? �� corresponds to a Fourier transform with support on the

positive�

axis and

? �� corresponds to a Fourier transform with support on the

negative�

axis.) Further, let )� �� � � � � � � 4 � � �? ������� � � � � with the same support

conventions as for � �� . Then we can write (5.27) in the form

� � ��� � ��� � � � � � � � � � )� �� � ��� � ��� �+� � �� � � � � � � � � � )� �� � � � � ��� �+� � �� � � � (5.30)

where� . � .3� � is the usual

* � � � � inner product, and

� � � ��� �+� � �� � � �&� �� 4 8 � � ��

�� � �

� � �while � ��� � � )� � � � � � � ��� � � � � � � � � � � � � � � � � � )� � � � � � � 9� � � � � � � � � � � � � 4� � )� � � � � � �� � � � � � ' �� , � � � � � # � ��� � � � � )� � � ��� � � ��� � �=��� �� )� � � � � � (5.31)

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where # � ��� � � 0 � � � � � � � � with � � � � � � � � � � . Comparing (5.31) with (1.9) wesee that

� � � � � )� � � � � � )� � � � � is just the echo generated at time � from the signal)� � � � � and the target position-velocity distribution # � ��� � � . Each inner product onthe right-hand side of (5.30) is the correlation function between the echo )� �� and atest signal

��� � ��� �+� � �� :

# � � � � �&� � � � � )� �� � � � � ��� �+� � �� � � � � )� �� � ��� � ��� �+� � �� � � � (5.32)

where � � � � � � �� 4 . Note that (5.32) provides a scheme for determining the

distribution # � � � � � experimentally: we can send out signals )� �� , measure theechos )� �� and then cross-correlate these echos with the test signals

�)� � ��� �+� � �� toconstruct # .

5.4 Exercises

1. Show that the representation� �

of the affine group � � is reducible.

2. Verify directly equation (5.15).

3. Verify directly equation (5.16).

4. Show that the rep� �

of � � is continuous in the sense that

� � � � � ��� � � � � � � � � � � � � � ��� � � � � � � � �for each � ( � 4 .

5. Show that the ambiguity and cross-ambiguity functions �� � � ��� �+� � � � �

are continuous functions of� ��� � � .

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Chapter 6

WEYL-HEISENBERG FRAMES

6.1 Windowed Fourier transforms

In this and the next chapter we introduce and study two procedures for the analysisof time-dependent signals, locally in both frequency and time. The first procedure,the “windowed Fourier transform” is associated with the Heisenberg group whilethe second, the “wavelet transform” is associated with the affine group.

Let �( * � � � � with � � �0� � � � and define the time-frequency translation of �

by�) � � � � * ����� � � ���� ;% � � ��� � � 4 � � � 4 � � 4 � � � � � � � ����� (6.1)

where � 4 is the unitary irred rep (4.5) of the Heisenberg group � � with� �

� . Now suppose � is centered about the point���%� � @���� in phase (time-frequency)

space, i.e., suppose� � � � � ����� � � � � ���� � � � @ � �� �A@ � � � � @$� @��where �� �A@ � � � � �

����� � � ���� 2! % � � is the Fourier transform of ������

. Then� � � � � ) � � � � * ����� � � � � ���� � � 4 � � � @ � �� ) � � � � * ����� � � � @$� @�� � � �so � ) � � � � * is centered about

��� � � � 4 � @�� � � � � in phase space. To analyze anarbitrary function

� �����in

* � � � � we compute the inner product

��� � 4 � � � � � � � � � ) � � � � * � � � � � ����� �� ) � � � � * ����� � �91

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with the idea that��� � 4 � � � � is sampling the behavior of

�in a neighborhood of

the point��� � � � 4 � @�� � � � � in phase space. As � 4 � � � range over all real numbers

the samples��� � 4 � � � � give us enough information to reconstruct

� �����. Indeed,

since � 4 is an irred rep of � � the functions � 4 � � 4 � � � � � � � � � ) � � � � * are dense in* � � � � as � � 4 � � � � runs over� � . Furthermore,

� ( * � � � � is uniquely determinedby the inner products

� � � � ) � � � � * � � � � � � 4 � � � � � . (Suppose� � 4 � � ) � � � � * � �� � � � � ) � � � � * � for

� 4 � � � ( * � � � � and all � 4 � � � . Then with� � � 4 � � � we have� � � � ) � � � � * �)0 �

, so�

is orthogonal to the closed subspace of* � � � � generated by

the � ) � � � � * . Since � 4 is irreducible this closed subspace is* � � � � . Hence

� � �and

� 4 � � � .)However, the set of basis states � ) � � � � * is overcomplete: the coefficients

� � � � ) � � � � * �are not independent of one another, i.e., in general there is no

� ( * � � � � such that� � � � ) � � � � * � � � � � 4 � � � � for an arbitrary� ( * � � � � � . The � ) � � � � * are examples

of coherent states, continuous overcomplete Hilbert space bases which are of in-terest in quantum optics, quantum field theory, group representation theory, etc.,[[69]].

As an important example we consider the case �� � � ����� � � � 4 8 < � � % 8 � ,

(4.27), the ground state. (Recall that �� � �

where � is the annihilation operatorfor bosons (4.19). This property uniquely determines the ground state.) Since

� �is essentially its own Fourier transform, (4.44), we see that �

� � �is centered

about��� � � @�� � � � � � � � in phase space. Thus

�) � � � � * �����&� � � 4 8 < � ���� ;% � � � '"% � � � , 8 � (6.2)

is centered about� � � 4 � � � � . It is very instructive to map these vectors in

* � � � �with inner product

� . � .3� to the Bargmann-Segal Hilbert space�

with inner product� . � . � , (4.32), via the unitary operator�

, (4.37). In�

the ground state is � � � � �&� � .Thus the corresponding coherent states are

�� ) � � � � * � � � � � 4 � � 4 � � � � � � � � �&� � ) � � � � *� � � �� ��� � � � � � 4 � � �� � � � ' � � � � ,� � � � � � � � �� ��� � � � � � 4 � � �� � � � � � 4 � � #" � � ' � � � � ,:8 � � � � � (6.3)

where � � � � � � ��� � �� � � ( �is the “delta function” with the property

� � � � � � �� � � � for each � ( �. Clearly�

� ) � � � � * � � ) � � � � * � � � � ) � � � � *� � � ) � � � � *� �� ��� � � � � � 4 � � �� � � �4 � � �� � � � � � 4 � � � � � 4 � � � ���� � ' � � � � , ' � � � � ,� � (6.4)

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so the � ) � � � * are not mutually orthogonal. Moreover, given� ( * � � � � with

� � � � ( �we have� � � � ) � � � � * � � � � � � ) � � � � *� �&� ��� � � � � � 4 � � �� � � � � � 4 � � � �

�� 4 � � � �� " � (6.5)

Expression (6.5) displays clearly the overcompleteness of the coherent states.Since � is an entire function, it is uniquely determined by its values in an openset of the complex plane (or a line segment or even on a discrete set of pointsin

�which have a limit point). Thus the values

� � � � ) � � � * � cannot be prescribedarbitrarily. However, from the “delta function” property

� � � ��� � � � � � �we can easily expand � ( �

as a double integral over the coherent states � ) � � � � *�,

hence we can expand� � � � 4 � ( * � � � � as the corresponding double integral

over the coherent states � ) � � � � * .There are two features of the foregoing discussion that are worth special em-

phasis. First there is the great flexibility in the coherent function approach dueto the fact that the function �

(�* � � � � can be chosen to fit the problem at hand.Second is the fact that coherent states are always overcomplete. Thus it isn’tnecessary to compute the inner products

� � � � ) � � � � * � � � � � 4 � � � � for every pointin phase space. In the windowed Fourier approach one typically samples

�at

the lattice points� � 4 � � � �*� � � ��� � � � where ��� � are fixed positive numbers and

� � � range over the integers. Here, ��� � and ������

must be chosen so that the map� � $ � � � ��� � � � ' is one-to-one; then

�can be recovered from the lattice point

values��� � ��� � � � .

Example 3 Given the function

������ � � � � � � �-, 4�� � � � � � 4� �

the set $�� ) ��� � * ' is an� �

basis for* � � � � � � � . Here, � � � run over the integers.

Thus � ) � � � � * is overcomplete.

6.2 The Weil-Brezin-Zak transform

The Weil-Brezin transform (earlier used in radar theory by Zak, so also calledthe Zak transform) (4.51) between the Schrodinger rep � 4 of � � on

* � � � � and

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the lattice rep)� , (4.48)-(4.50), is very useful in studying the lattice sampling

problem, particularly in the case �� � � � . Restricting to this case for the time

being, we let� ( * � � � � . Then

� � � � 4 � � � �&� � � � � 4 � � � � � � �

� � ���� � � � � 4 � �

(6.6)

satisfies� � �� 4 � � 4 � � � � � �&� � � ���& � � � � � � 4 � � � �

for integers 4 � � . (Here (6.6) is meaningful if

�belongs to, say, the Schwartz

class. Otherwise � � � � � � � � : � �� where

� � � ��� � : � � and the�� are

Schwartz class functions. The limit is taken with respect to the Hilbert spacenorm.) Furthermore

� � 4 � � 4 � � � � � � � � � � � 4 � � � � � � � )� � � 4 � � � � � � � � � � 4 � � � �� ��� � " � � � 4 � � � � � � � 4 � � 4 � � � � � � �� Hence if

� �� ) �&� � * � � 4 � � � � � � we have

�) �&� � *� � � 4 � � � �&� ��� � " � � � � 4 � � � � � � � � � � � 4 � � � �� (6.7)

Thus in the lattice rep, the functions �) ��� � *�

differ from ��

simply by the multi-plicative factor � ���� 5' � � � � � � , � � � � � � � 4 � � � � , and as � � � range over the integersthe � � � � form an

� �basis for the Hilbert space of the lattice rep:� � 4 � � � �&� � 4� � 4� � 4 � � 4 � � � � � � � � 4 � � � � � � 4 � � � (6.8)

Theorem 17 For� ��� � �,� � � � � � and �

( * � � � � the transforms $�� ) �&� � * % � � � ���� � � � " � . . . ' span* � � � � if and only if � �

� � 4 � � � � � � � �� � � 4 � � � � �� �

a.e..

PROOF: Let � be the closed linear subspace of* � � � � spanned by the $�� ) �&� � * ' .

Clearly � � * � � � � iff� � �

a.e. is the only solution of� � � � ) �&� � * � � �

for allintegers � and � . Applying the Weyl-Brezin -Zak isomorphism � we have� � � � ) �&� � * � � � � � � � � � � � �

�� � � � � � � � � � � � � � � � � � ��� �� � � � � � � �� (6.9)

Since the functions � � � � form an���

basis for the Hilbert space (6.8) it followsthat

� � � � ) ��� � * � � �for all integers � � � iff

��� � � 4 � � � � � � � � 4 � � � � � �, a.e.. If

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�� �� �

, a.e. then� � � � � �

and � � * � � � � . If �� � �

on a set

?of positive

measure on the unit square, then the characteristic function��� � � � ���

satisfies���

�� ���

� �� � �

a.e., hence� � � � ) �&� � * � � �

and � �� * � � � � . Q.E.D.In the case �

����� � � � 4 8 < � � % 8 � one finds that

�� � � 4 � � � ��� � � 4 8 <

� � ���� � � ' � � � , 8 � (6.10)

As is well-known, [[40]], [[110]], the series (6.10) defines a Jacobi Theta function.Using complex variable techniques it can be shown that this function vanishes atthe single point

� 4� � 4� � in the square� , � 4 � � , � , � � � � , [[110]]. Thus

�� �� �

a.e. and the functions $�� ) ��� � * ' span* � � � � . (However, the expansion of

an* � � � � function in terms of this set is not unique and the $�� ) �&� � * ' do not form a

frame in the sense of � 6.4.)

Corollary 6 For� � � � � � � � � � � and �

( * � � � � the transforms $�� ) �&� � * % � � � �� � � � � . . . ' form an���

basis for* � � � � iff � � � � � 4 � � � � � � � , a.e.

PROOF: We have-� � - ��� � �

� ) �&� � * � � ) � � � * � � ���� � � � � � � � � � � � �� � � � � � � � � � � � � � � � �

iff � � � � � � � , a.e. Q.E.D.As an example, let �

��� ) � � 4 , where

� ) � � 4 , ������� � � ��� � , �� �� ��� ��� � � � � �

Then it is easy to see that � � � � � 4 � � � � ��0 � . Thus $�� ) �&� � * ' is an� �

basis for* � � � � .Theorem 18 For

� � � � � � � � � � � and �( * � � � � , suppose there are constants9 ��� such that �

� 9 , � � � � � 4 � � � � � � , � � �almost everywhere in the square

� , � 4 � � � � � . Then $�� ) ��� � * ' is a basisfor

* � � � � , i.e., each� ( * � � � � can be expanded uniquely in the form

� ����� � � � � � ) �&� � * . Indeed,

� � �� � ��� � � ) �&� � *� � � � � � � � � ��� � � � � � � � �

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PROOF: By hypothesis � � � � � 4 is a bounded function on the domain� , � 4 � � � �� . Hence

��� � � is square integrable on this domain and, from the periodic-ity properties of elements in the lattice Hilbert space,

���� � � � 4 � � � � � � � � �

���� � � � 4 � � � � . It follows that

���

��

� � � ��� � �

where � � �� � ��� � � � � � � � � , so

��� � � � � ��� � � � � . This last expression im-

plies� � � � � � � ) ��� � * . Conversely, given

� � � � � � � ) �&� � * we can reverse thesteps in the preceding argument to obtain � � �

� � ��� � � � � � � � � . Q.E.D.

6.3 Windowed transforms and ambiguity functions

We can relate these expansions to radar cross-ambiguity functions as follows. Theexpansion

� � � � � � � ) � � � * is equivalent to the lattice Hilbert space expansion��� � � � � �

�� � � � � or

��� �� � � � � � � � � � � � � � � � � (6.11)

Now if ��

is a bounded function then� � �� � � � 4 � � � � and � � � � � both belong to the

lattice Hilbert space and are periodic functions in � 4 and � � with period � . Hence,��� �� � � � � � �

�� � �� � � � � � � � � � � � � �

with� � �

� � ��� �� � � � � � � �&� � ��� � � � � � � � �&� � � � � ) ��� � * � � � � � � 4 � � � � � � � �� � � � ��� �� � � � � � � � � �

� � � ) �&� � * � � �� � � 4 � � � � � � �

Thus (6.11) gives the Fourier series expansion for� � �� � as the product of two

other Fourier series expansions. (We consider the functions� � � , hence

� � � � � asknown.) The Fourier coefficients in the expansions of

� � �� � and � � � � � are cross-ambiguity functions. If � � � � � never vanishes we can solve for the � � � directly:

� � ��� � � � � � � �

�� � � �'� � �� � � � � � ��

where the � �� � are the Fourier coefficients of � � � � � � . However, if � � � � � vanishes atsome point then the best we can do is obtain the convolution equations �

� ��� � ,i.e.,

� � �� � ��� � � � � � �� � � � (6.12)

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(Auslander and Tolimieri [[10]] have shown how to approximate the coefficients� � even in the cases where � � � � � vanishes at some points. The basic idea is totruncate

� � � ��� � � to a finite number of nonzero terms and to sample equation

(6.11), making sure that � � � � � � 4 � � � � is nonzero at each sample point. The � � �can then be computed by using the inverse finite Fourier transform (2.37).)

The problem of � � � � vanishing at a point is not confined to an isolated example,such as (6.10). Indeed it can be shown that if �

is an everywhere continuousfunction in the lattice Hilbert space then it must vanish at at least one point, [[55]].

6.4 Frames

To understand the nature of the complete sets $�� ) �&� � * ' it is useful to broaden ourperspective and introduce the idea of a frame in an arbitrary Hilbert space

�.

In this more general point of view we are given a sequence $ � � ' of elements of�

and we want to find conditions on $ � � ' so that we can recover an arbitrary� ( �

from the inner products�� � � � � on

�. Let

* � � � � be the Hilbert space ofcountable sequences $ � � ' with inner product

� � � � � � �� � � �� � . (A sequence $ � � '

belongs to* � � � � provided

�� � � �� � ��� .) Now let � % � � * � � � � be the linear

mapping defined by � � � � � � �� � � � � (6.13)

We require that � is a bounded operator from�

to* � � � � , i.e., that there is a finite� � �

such that�� � � � � � � � � � , � � � � � � � . In order to recover � from the

�� � � � � we

want � to be invertible with � � 4 %���� � �where

���is the range � � of� in

* � � � � . Moreover, for numerical stability in the computation of � from the�� � � � � we want � � 4 to be bounded. (In other words we want to require that a

“small” change in the data�� � � � � leads to a “small” change in � .) This means that

there is a finite 9 � �such that

�� � � � � � � � � � � 9!� � � � � � . (Note that � � 4 � � � if

� � � �� � � � � .) If these conditions are satisfied, i.e., if there exist positive constants9 ��� such that 9�� � � � � � ,

�� � � � � � � � � , � � � � � � � (6.14)

for all � (��, we say that the sequence $ � � ' is a frame for

�and that 9 and �

are frame bounds.The adjoint � � of � is the linear mapping � � %-* � � � � � �

defined by� � � � � �/� � � � � � � �97

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for all � (+* � � � � , � ( �. A simple computation yields� � � �

�� � � � (6.15)

(Since � is bounded, so is � � and the right-hand side of (6.15) is well-defined forall � ( * � � � � .) Now the bounded self-adjoint operator

� � � � � % � � �is

given by � � � � � � � � �

�� � � � � � � � (6.16)

and we can rewrite the defining inequality (6.14) for the frame as9!� � � � � � , � � � � � � �/� , � � � � � � � (6.17)

Since 9 � �, if � � � � � then � � , so

�is one-to-one, hence invertible.

Furthermore, the range� �

of�

is�

. Indeed, if� �

is a proper subspace of�

then we can find a nonzero vector � in� � � � � % � � � � � � � �

for all � ( �.

However,� � � � � � � � � � � � � � � � � � � � � � �*� �

��� � � � � � � � ��� � . Setting � � �

we obtain �� � � � � � � � � � �

By (6.14) we have � � , a contradiction. thus� � � �

and the inverse operator� � 4 exists and has domain�

.Since

� � � 4 � � � � 4 � � � � for all � ( �, we immediately obtain two expan-

sions for � from (6.16):� � � � �

�� � � 4 � � � � � � � � � �

�� � � � 4 � � � � �

� � � � ���� � � � � � � 4 � � (6.18)

(The second equality in (6.18) follows from the identity� � � 4 � � � � � � �

� � � � 4 � � � ,which holds since

� � 4 is self-adjoint.)Recall that for a positive operator

�, i.e., an operator such that

� � � � �/� � � forall � (��

the inequalities 9!� � � � � � , � � � � � � , � � � � � � �for 9 ��� � � are equivalent to the inequalities9�� � � � �-, � � � � � � , � � � � � � � (6.19)

see [[95]] or [[38]].An examination of equations (6.18) suggests that if the $ � � ' form a frame then

so do the $ � � 4 � � ' .98

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Theorem 19 Suppose $ � � ' is a frame with frame bounds 9 ��� and let� � � � � .

Then $ � � 4 � � ' is also a frame, called the dual frame of $ � � ' , with frame bounds� � 4 � 9 � 4 .PROOF: Setting � � � � 4 � in (7.19) we have � � 4 � � � � ��, � � � � 4 � � ��, 9 � 4 � � � � � .Since

� � 4 is self-adjoint, this implies � � 4 � � � � � � , � � � 4 � ��� �;, 9 � 4 � � � � � � . Fromthe last equation (6.18) we have

� � 4 � � ��� � � 4 � � � � � � � 4 � � so

� � � 4 � ��� � ���� � � 4 � � � � � � � � 4 � � ��� � � �

� � � � � � � 4 � � � � � . Hence $ � � 4 � � ' is a frame withframe bounds � � 4 � 9 � 4 . Q.E.D.

We say that $ � � ' is a tight frame if 9 � � .

Corollary 7 If $ � � ' is a tight frame then every � ( �can be expanded in the

form� � 9 � 4

�� � � � � � �

PROOF: Since $ � � ' is a tight frame we have 9�� � � � � � � � � � � �/� or� � � � 9 � � � � �/� ��

where � is the identity operator � � � � . Since� ��9 � is a self-adjoint operator

we have � � � � ��9 � � � � � � �for all � (��

. Thus� � 9 � . However, from (6.18),� � � � �

�� � � � � � � . Q.E.D.

6.5 Frames of� � �

type

We can now relate frames with the Heisenberg group lattice construction (6.6).

Theorem 20 For� ��� � � � � � � � � and �

( * � � � � , we have�� 9 , � � � � � 4 � � � � � � , � � � (6.20)

almost everywhere in the square� , � 4 � � � � � iff $�� ) ��� � * ' is a frame for* � � � � with frame bounds 9 ��� . (By Theorem 18 this frame is actually a basis

for* � � � � .).

PROOF: If (6.20) holds then ��

is a bounded function on the square. Hence forany

� (+* � � � � � ��� �� � is a periodic function, in � 4 � � � on the square. Thus� ��� � � � � � � � ) �&� � * � � � � �

�&� � � � � ��� � � � � � � � � � �� � �&� � � � � ��� �� � � � � � � � � � � � � ��� �� � � � �� � 4� � 4� � ��� � � � � � � � � � 4 � � � (6.21)

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(Here we have used the Plancherel theorem for the exponentials�� � � ) It follows

from (6.20) that

9�� � � � � � , ��� � � � � � � � ) �&� � * � � � , � � � � � � � � (6.22)

so $�� ) �&� � * ' is a frame.Conversely, if $�� ) �&� � * ' is a frame with frame bounds 9 ��� , it follows from

(6.22) and the computation (6.21) that9!� � ��� � � � , � 4� � 4� � ��� � � � � � � � � � 4 � � � , � � � ��� � � �for an arbitrary

���in the lattice Hilbert space. (Here we have used the fact that� � � � � � � � ��� � � , since � is a unitary transformation.) Thus the inequalities (6.20)

hold almost everywhere. Q.E.D.Frames of the form $�� ) � � � � � * ' are called Weyl-Heisenberg (or W-H) frames.

The Weyl-Brezin-Zak transform is not so useful for the study of W-H frames withgeneral frame parameters

� � � � � . (Note from (6.1) that it is only the product � � thatis of significance for the W-H frame parameters. Indeed, the change of variable� � � � � in (6.1) converts the frame parameters

� � � � � to� � � � � � ��� � � � � � � .) An

easy consequence of the general definition of frames is the following:

Theorem 21 Let �( * � � � � and ��� � � 9 ��� � � such that

1.�� 9 , �

� � � � � � � � � � � , � � � , a.e.,

2. � has support contained in an interval5

where5

has length �� 4 .

Then the $�� ) � � � � � * ' are a W-H frame for* � � � � with frame bounds �

� 4 9 � � � 4 � .

PROOF: For fixed � and arbitrary� ( * � � � � the function

������� � � �����

���� � � � �

has support in the interval5�� $ � � � � % � ( 5 '

of length �� 4 . Thus

�������

can beexpanded in a Fourier series with respect to the basis exponentials

���������� � ���� � � %

on5� . Using the Plancherel formula for this expansion we have��&� � � � � � � ) � � � � � * � � � � �

�&� � � � � � � � �� � � �� 4

��� � � � � � � �)� � � 4

������� � � ����� � � � � ��� � � � � � � � �� 4

�� � � � ����� � � � � � � ��� � � � � � � � ��

100

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From property 1) we have then9� � � � � � � ,

�&� � � � � � � ) � ��� � � * � � � , �� � � � � � � �

so $�� ) � � � � � * ' is a W-H frame. Q.E.D.There are no W-H frames with frame parameters

� ��� � � such that � � � � ,[[14]], [[94]]. For some insight into this case we consider the example

� ��� � � �� � � � � � � � � , � an integer. Let �( * � � � � . There are two distinct possibilities:

1. There is a constant 9 � � such that 9 , � � � � � 4 � � � � � almost everywhere.

2. There is no such 9 � � .Let � be the closed subspace of

* � � � � spanned by the functions $�� ) � �0� � * � � � � ���� � � � " � . . . ' and suppose� ( * � � � � . Then� � � � ) � �0� � * � � � ��� � � � � � � � � ��� � ��� �� � � � � � � � ��

If possibility 1) holds, we set� � � �� � 4� �

� # � 4 . Then���

belongs to the latticeHilbert space and

� � �$�� # � 4 � � � � � � � � � ��� �� � � � � � � ��� � � � � ) � � � � * � so

� ( � �

and $�� ) � � � � * ' is not a frame. Now suppose possibility 2) holds. Then accordingto the proof of Theorem 7.4, � cannot generate a frame $�� ) �&� � * ' with frame pa-rameters

� � � � � because there is no 9 � �such that 9�� � � � � � � � �&� � � � � � � ) �&� � * � � � .

Since the $�� ) � � � � * ' corresponding to frame parameters� � � � � is a proper subset

of $�� ) �&� � * ' , it follows that $�� ) � � � � * ' cannot be a frame either.For frame parameters

� ��� � � with�� � � � � it is not difficult to construct W-H

frames $�� ) � � � � � * ' such that �( * � � � � is a smooth function [[33]], [[56]], [[55]].

Taking the case �� � � � � 4� , for example, let � be an infinitely differentiable

function on�

such that � � � �&� � � ��� � , �� ��� � � �

and�� � � � � � � if

�� � � � . Let7 � � � � � ��� � � � � � � � � �

and set

�� � � �

����� ����� � � , �� � � � � �

� � � �7 � � � � � , � , "� � " � � (6.23)

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Then �( * � � � � is infinitely differentiable and with support contained in the in-

terval � � � " � . Moreover, � � �0� � � � � and�� � � � � � � � � � 0/� . It follows immediately

from Theorem 21 that $�� ) ��� � 8 � * ' is a W-H frame with frame bounds 9 � � � " .We conclude this section by deriving some identities related to cross-ambiguity

functions evaluated on lattices of the Heisenberg group.

Theorem 22 [[97]], [[98]] Let� � � ( * � � � � such that � ��� � � 4 � � � � �0� � � � � 4 � � � � �

are bounded almost everywhere. Then ��� � � � � � � ) ��� � * � � � �

��� � � � � � ) ��� � * � � � ) �&� � * � � � PROOF: Since

� � � � ) �&� � * � � � ��� � � � � � � � � � � ��� �� � � � � � � � we have the Fourierseries expansion

��� � � 4 � � � � � � � � 4 � � � � � ��� � � � � � ) �&� � * � � � � � � � 4 � � � �� (6.24)

Since � ��� � � � � � � are bounded,��� �� � is square integrable with respect to the measure� � 4 � � � on the square

� , � 4 � � � � � . From the Plancherel formula for doubleFourier series, we obtain the identity� 4� � 4� � ��� � � � � � � � � � 4 � � � �

��� � � � � � � ) �&� � * � � � (6.25)

Similarly, we can obtain expansions of the form (6.25) for� � ����

and �� �� � . Ap-

plying the Plancherel formula to these two functions we find� 4� � 4� � ��� � � � � � � � � 4 � � � � �&� � � � � � ) �&� � * � � � ) ��� � * � ��� (6.26)

Q.E.D.

6.6 Exercises

1. Verify that if �( * � � � � , � � �0� � � � and � is centered about

���%� � @�� � in phasespace, then � ) � � � � * is centered about

��� � � � 4 � @�� � � � � .2. Given the function

������ � � � � � � �-, 4�� � � � � � 4� �

show that the set $�� ) �&� � * ' is an���

basis for* � � � � .

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Chapter 7

AFFINE FRAMES ANDWAVELETS

7.1 Continuous wavelets

Here we work out the analog for the affine group of the Weyl-Heisenberg framefor the Heisenberg group. Let �

( * � � � � with � � �0� � � � and define the affinetranslation of � by

� ' ��� � , ������� �� 4 8 � �

� � � �� � ����� � ��� �+� � ����� (7.1)

where � � �and

� �is the unitary rep (5.10) of the affine group. Recall that��� - � � � � � is reducible. Indeed

* � � � �%� � ��� �

where� �

con-sists of the functions

� � such that the Fourier transform� � � � � � has support on

the positive � -axis and the functions� � in

� �have Fourier transform with sup-

port on the negative � -axis. Thus the functions $�� ' ��� � , ' will not necessarily span* � � � � . However, if we choose two functions �� ( � �

with � � � � � � � � thenthe functions $�� ' � � � ,� � � ' ��� � ,� % � � � '

will span* � � � � . By translation in

�if nec-

essary, we can assume that� � � � � � ����� � � � � � �

. Let � � � � �0� � � � � � � � � � � , � � �

�� �0� � � � � � � � � � � . Then ��

are centered about the origin in position spaceand about

�in momentum space. It follows that� � � � � ' � � � ,� ����� � � � � � � � � �

� � �0� � � ' � � � ,� � � � � � � � � � �� 4 �

To define a lattice in the affine group space we choose two nonzero real numbers� � � � � � �

with �� �� � . Then the lattice points are �

� � � � � � � � �� � � � , � � � �

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� � � � � . . . , so

�� � �������

� ' ���# � � � # � �# , �����&� �� � 8 ��

�� � � �� � � � �

���� (7.2)

Thus � ���

is centered about � � � � � � � in position space and about �� �� �

in mo-mentum space. Note that if � has support contained in an interval of length �

then the support of � ��

is contained in an interval of length �� �� � . Similarly, if�

� has support contained in an interval of length*

then the support of�� �

�is

contained in an interval of length � �� *

. (Note that this behavior is very differentfrom the behavior of the Heisenberg translates � ) � ��� � � * . In the Heisenberg case thesupport of � in either position or momentum space is the same as the support of� ) � ��� � � * . In the affine case the sampling of position-momentum space is on a loga-rithmic scale. There is the possibility, through the choice of � and � , of samplingin smaller and smaller neighborhoods of a fixed point in position space, [[26]],[[32]].)

The affine translates � ' � � � ,� are called (continuous) wavelets and each of thefunctions �

is a mother wavelet. The map � % � � � � � � � �� � is the wavelettransform

NOTE: There is another way to get around the fact that the representation��� - � � � � � is reducible, and that avoids using two continuous wavelets. Let�( * � � � � with � � �0� � � � and define the (extended) affine translations of � by

� ' ��� � , �����&� � � � � 4 8 � �� � � �� � � � � � � � � � � � ����� (7.3)

where � �� �. Here

� � �defines a unitary rep of the extended affine group � � � . This

is the matrix group with elements

9 � ��� � ��� �� �� � � � � �� �

(7.4)

and multiplication rule � ��� � �'� � � � � � � � � � � � � � � � � � �� Here, � � � contains � � as the subgroup of elements with � � �

. As a manifoldit has two connected components � � � � � � � �

�� where �

�� is the set (not a

subgroup) of elements of � � � with � ��.

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Theorem 23 Let� � �

be the representation of � � � on* � � � � defined by operators� � � � ��� �+� such that

� � � � ��� �+� � ����� � � � � � 4 8 � �� � � �� �

for any� �����)( * � � � � and 9 � ��� � � ( � � � . Then

� � �is a unitary irred rep of � � � .

The proof is almost immediate from the proof of Theorem 16. Indeed theinduced action of the operators

� � � � ��� �+� on the space of Fourier transforms �� � � �agrees with the action of the operators

� � � ��� �+� in the proof of that theorem, exceptthat here � can take negative values and the functions �� � � � are defined on the realaxis, not just the half-axis � � � .

It follows from this theorem the functions $�� ' ��� � , ' for any � with � � � � � �� �will necessarily span

* � � � � , as� ��� � � runs over � � � , although if we restrict to the

subgroup � � spanning may not be possible:

� � � � ��� - � � � � � 7.2 Affine frames

The general definitions and analysis of frames presented in Chapter 6 clearly applyto wavelets. However, there is no affine analog of the Weil-Brezin-Zak transformwhich was so useful for Weyl-Heisenberg frames. Nonetheless we can prove thefollowing result directly.

Lemma 4 [[33]] Let �( * � � � � such that the support of

�� is contained in the

interval � ��� * � where�� � �

*� � , and let �

� � � � � � � � with�$* � � � � � , � .

Suppose also that �� 9 ,

�� � �

� � � � � � � � , � ���

for almost all � � � . Then $�� � �(' is a frame for� �

with frame bounds 9 � � � � � � .PROOF: The demonstration is analogous to that of Theorem 21. Let

� ( � �and

note that �( � �

. For fixed � the support of� � � � � � � � � �

� � � is contained in the

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interval � , � , ��� � � � (of length � � � ). Then��&� � � � � � � � � � � � � �

��� � � � � � � � � �� � � �� �

��� � � � �� � � � � � � � � � �� � � � �� � � � � � � # � � � � �� ��� � �

� ����� � � ��� � � ��� � � � �� ���#� # � � � 4 8 � #� � � � � � � �� � � � �

� � � � � � �� 4���� � � � � � � � � �

� � � � � � � � �=�� 4�� � � � � � � � � � � � � � � �

� � � � � � � � � � � Since � � � � � � � � � � � � � � � � � � � for

� ( � �, the result9�� � � � � � ,

�&� � � � � � � � � � � � , � � � � � � �follows. Q.E.D.

A very similar result characterizes a frame for� �

. (Just let � run from � �to�.) Furthermore, if $�� � �� ' � $�� � �� '

are frames for� � � � �

, respectively, corre-sponding to lattice parameters �

� � � � , then $�� � �� � � � �� 'is a frame for

* � � � �Examples 3 1. For lattice parameters �

� � " � � � � � , choose � � �.� ) 4 � � , and� � � � ' � � � � 4 * . Then � � generates a tight frame for

� �with 9 � � � �

and � � generates a tight frame for� �

with 9 � � � � . Thus $�� � �� � � � �� 'is a tight frame for

* � � � � . (Indeed, one can verify directly that $�� � �� 'is an���

basis for* � � � � .

2. Let � be the function such that

��� � � � �� �

� �

������� ������� � � � , �� � � � � � � � � �� ' � � 4 , � ��� � � � , � �� � � � � � � � � � �� � ' � � 4 , � ��� � � � � , � � �� � � � � � � �

where � � � � is defined as in (6.23). Then $�� � �+' is a tight frame for� �

with9 � � � 4��� � �

. Furthermore, if � � � � and � � � �� then $�� � �� 'is a tight

frame for* � � � � .

Suppose �( * � � � � such that

��� � � is bounded almost everywhere and has

support in the interval � � 4� � � 4� � � . Then for any� ( * � � � � the function

�� � 8 �� � � � � � �� � � � �

� � �106

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has support in this same interval and is square integrable. Thus���� � � � � � � � � � � � � �

��� � � � � � 8 �� � � � � � � � �� � � � �� � � � � ���& � � # � �0� �� �

� 4� ��� � � � �� � � � � � � �� � � � �

� � � � � � �� 4� # �

�� � � � � � � � � � � � � �� � � � � � � � �=�

� 4� # � � � � � � � � � � � � � � �

� � � � � � � � � � It follows from the computation that if there exist constants 9 ��� � � such that9 ,

�� � �

� � � � � � � � , �

for almost all � , then the single mother wavelet � generates an affine frame.We conclude this section with two examples of wavelets whose properties do

not follow directly from the preceding theory. The first is the Haar basis generatedby the mother wavelet

������ � ����� ����

� � � � � �� ��� � , �� 4���� � � 4� , � , �� ���>� � � �

where �� " � � � � . One can check directly that $�� � � ' is not only a frame, it is

an� �

basis for* � � � � .

The Haar wavelets have discontinuities. However, Y. Meyer discovered an���basis for

* � � � � whose mother wavelet � is an infinitely differential functionsuch that

�� has compact support. The lattice is �

� " � � � � . The Meyer waveletis defined by

��� � � � � �+8 � @ � � �0� � where

@ � � � � � ������� �����

� ��� � , 4�� ��� � � � ��� � � � � ��� 4� , � , ��� � � � � � � � �� � � � � � � � , � , <�� ��� < � , �

and � is defined as in (6.23), except that in addition we require � � � � � � � � � � � �� for� , � , � . One can check that � � �0� � � � � and

�� � � �

� " � � � � � � � .Moreover, it can be shown that � generates a tight frame for

* � � � � with framebounds 9 � � � � , and, indeed, that $�� � �(' is an

���basis for

* � � � � . ThenIngrid Daudechies, in 1989, went on to find an infinite family of

� �bases for* � � � � whose mother wavelets have differentiability of arbitrarily high order.

A theory which “explains” the orthogonality found in these last examples ismultiresolution analysis [[55]], [[73]], [[30]]; it is the subject of another set ofnotes.

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7.3 Exercises

1. Suppose �( * � � � � with � � � � � � � and � is centered about

� � � � in theposition-momentum space. Show that � ' � � � , is centered about

� � � � � � 4 � .2. Prove directly that the Haar basis is an

���basis for

* � � � � .3. For �

�����&� � � % , show that the functions � ' � � � , are dense in* � � � � .

108

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Chapter 8

THE SCHRODINGER GROUP

8.1 Automorphisms of� �

We have already seen that the infinite-dimensional irred unitary reps � � of theHeisenberg group � � extend naturally to irred reps of the four-parameter oscil-lator group and that a study of the oscillator group reps provides insight into thebehavior of the � � reps, � 4.7. In fact, the � � reps extend to reps of the six-parameter Schrodinger group. An understanding of the action of the Schrodingergroup provides an explanation for a number of the “deep” transformation proper-ties of objects such as radar ambiguity functions and Jacobi Theta functions.

We start by searching for automorphisms of ��� , i.e. one-to-one maps�

of� � onto itself such that

� � 9 � ��� � � 9 � � � � � for 9 � � ( ��� . Using the usualcoordinate representation

9 � ��� � � � � � ��� � � �� � �� � �� � (8.1)

for � � , so that the group product is9 � ��� � � � � 9 � � � � � � � � � � � 9 � � � � � � � � � � � � � � � � � � � � �we can write

� � 9 �'� ��� � � � �&� 9 � � 4 � ��� � � � � � � � � ��� � � � � � � � � ��� � � � ��� (8.2)

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where� � � 4 � � � � � � � � � � � � � � � � � � � � � � 4 � ��� � � � � � � 4 � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � �

� � 4 � ��� � � � � � � � � � � � � � � � �� (8.3)

Under the assumption that the� � are continuously differentiable functions, we

shall determine all such automorphisms.Before proceeding with this task, let us see why it could be relevant to radar

and sonar. The radar cross-ambiguity function takes the form��� ��� � � � � � � � 4 � ��� � � � � � � � �

(up to a harmless exponential factor arising from the � coordinate) where� � � (* � � � � and � 4 � ��� � � � � � � 4 � 9 � ��� � � � � � is the irred rep (4.5) of � � . If�

isan automorphism of ��� then � 4 � � � 9 � � ��� � � � ��� � � 4� � ��� � � � � also defines anirred unitary rep of � � on

* � � � � . (A rep since�

preserves the group mul-tiplication property and irreducible since the operators � 4� are just a reorder-ing of the operators � 4 .) Thus � 4� is equivalent, hence unitary equivalent, toone of the standard unitary irred reps � � of � � that we have already studied.Suppose this rep is � 4 itself, i.e., suppose there is a unitary operator � suchthat � 4� � ��� � � � � � �

� 4 � 4 � ��� � � � � � . Then� � 4 � � � 9 ��� � � � � � � � 4� � ��� � � � � � � � � ��

�� 4 � 4 � ��� � � � � � � � � � � � � 4 � ��� � � � � � � � � ��� � � ��� ��� � � � � , which is again an am-

biguity function. Thus if� � ��� � � � � is an ambiguity function, then so is

��� � 4 � � � � � � � � � � � � � � ��� .Now we compute the possible automorphisms

�. Differentiating (8.3a) with

respect to ��we find

� 4 � 4 � � � � � � � � � � � � � � � � � � � �&� � 4 � 4 � � � � � � � � � �� Since the right-hand side of this equation is independent of ��� � � � , we have

� 4 � 4 � ��� � � � �� 4 , a constant. Similarly, differentiating (8.3a) with respect to � we have� � � 4 � ��� � � � �&� 0 . Differentiating (8.3a) with respect to � yields4 � � � � � � 4 � � � � � � � � � � � � � � � � � � � �&� 4 Since this equation holds for general �

� � � � � � � , it follows that� � � 4 � ��� � � � ��� �

.Thus

� 4 � ��� � � � � � 4 � � 0�� � where

is a constant. Substituting this expressionback into (8.3a) we see that

�� �. Similarly, equation (8.3b) has only the solution� � � ��� � � � ��� � � � - � where � � - are constants. The computation for equation

(8.3c) is just as straightforward, although the details are a bit more complicated.The final result is� 4 � ��� � � � � � 4 � � 0�� � � � � ��� � � � � � � � � - � �� � � ��� � � � � � � � � � � � 4� � 4 � � 0�� �'� � � � - � � � � 4 - �10 � �'� � � 4� � � �� (8.4)

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Here, 4 � 0 � � � - � � � � are real constants such that 4 - �10 � �� �, so that

�is 1-1.

Some of the automorphisms of ��� are inner automorphisms. These are theautomorphisms of the form

��� 9 � � � � 4 9 � for 9 (

� � , where � is a fixedmember of � � . (Clearly,

�� maps � � onto itself and is one-to-one. Furthermore�

�� 9 4 9 � � � � � 4 9 4 9 � � � � � � 4 9 4 � � � � � 4 9 � � � � � � � 9 4 � � � � 9 � � , so

�� is a

group homomorphism.) If � � � � � � � � � � � � � then

��� 9 �&� � � 4 9 � ��� � � � � � � ��� � � ��� � � � � � � � � �� � � � �� � � � �

��� 9 � ��� � � � � � � � � � � � �so the transformations

� 4 � ��� � � � � � � � � � � ��� � � � � � �� � � ��� � � � � � � � � � � � � � � (8.5)

correspond to inner automorphisms. We are not very interested in inner automor-phisms because they can easily be understood in terms of the Heisenberg groupitself. Thus we set �

� � � �in (8.4) and concentrate on the outer automor-

phisms� 4 � ��� � � � � � 4 � � 0�� � � � � ��� � � � � � � � � - � �� � � ��� � � � � � 4� � 4 � � 0 � �'� � � � - � � � � 4 - � 0 � � � � � 4� � � �� (8.6)

Recall that the infinite-dimensional irred unitary reps � � of � � take the form,(4.5), � � � ��� � � � � � �����&� � ���� � ' � � % �+, � ��� � � �for � ( * � � � � , where

�is a nonzero real constant. Note that the operators

corresponding to the center�

of ��� are just multiples of the identity operator:� � � � � � � � � � � ���� � � � . Since � �� � � � � � � � � � ���� � ' 2 � � 6 � , � � the irred rep � �� can pos-sibly be equivalent to � � only if 4 - � 0 � � � , so we now restrict our attention tothis case.

With this restriction � �� must be equivalent to � � . Indeed, the reps � � � � �� co-incide on the center

�. Furthermore, the matrix elements � � � � ��� � � � � are square

integrable with respect to the measure � � �=� in the plane. (In fact, these ma-trix elements differ from �� � ��� � � � � only by a factor of absolute value � and achange of variables with Jacobian 4 - � 0 � � � .) Thus, if � �� is not equiva-lent to � � we can use Corollary 1 and repeat the arguments leading to (4.8) for

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� ' � , 0 � � � � ' � , 0 � �� � � �� �, and measure � � � � to obtain� � � � �� � � � � ��� � � � � �: � ��� � � � � � � � � � �

for all ����� � � . Thus the matrix elements of � �� are orthogonal to those of � �in

* � � � � � . However, as we have shown in � 4.6, the matrix elements �� � ��� � � � �form a basis for

* � � � � � . This contradiction proves that � �� � � � � , hence that thereexist unitary operators �

� ��� � � such that

� �� � ��� � � � � ��� 4� � � � ��� � � � � � �� � � � 4 � � 0�� � � � � - � � 4� � 4 � � 0�� �'� � � � - � � � � � 4� � � �� ��� � � � � ��� 4 � � 0�� �'� � � � - � � � � � � � � � �&4 � � 0 � � � � � - � � � �

(8.7)

where # � � 4 0� - � and � � # � 4 - �10 � � � .

Theorem 24 Suppose� � � ��� � � � � is a matrix element of the irred unitary rep � �

of � � . Then

���� ��� � � � � � ��� � � � � � � 4 � � 0�� � � � � � - � � � � � � � �� � � 4 � � 0�� � � � � - � � � � (8.8)

is also a matrix element of � � for any # � � 4 0� - � with ��� # � 4 - � 0 � � � .

PROOF: Suppose� � � ��� � � � � � � � � � ��� � � � � � 4 � � � � for � 4 � � � ( * � � � � . Setting� � � � � � � � � � " , we obtain (8.8) from (8.7) with �

� � ��� � � � �&� � � � � ��� � � � � � 4 � � � � .Q.E.D.

Note that (8.8) gives us information about the structure of the set of am-biguity and of cross-ambiguity functions.

8.2 The metaplectic representation

Next we turn to the problem of actually computing the operators � � . First ofall, note from (8.7) that for any phase factor � � ' � , (with � � � � � � ) the unitaryoperators �

��� � � � � also satisfy (8.7). Indeed, a simple argument using Theo-

rem 12 shows that the operators � � are uniquely determined up to a phase factor.

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We shall find that it is possible to choose the operators � � such that the mapping� �

� � � is continuous in the norm as a function of the local parameters 4 � 0 � � � -for every � ( * � � � � .

It is no accident that we have arranged the parameters 4 � 0 � � � - in the form ofthe matrix # ( ? * � " � � � , since � � # � � . Indeed, it is straightforward to checkthat if � �� � ��� � �

� 4� � � � � � � � � � �� � � � � �� 4� � � ��� � �

for automorphisms�

and� �

of � � , then the automorphism� � � % � � � � � � � � � �

corresponds to the matrix # # � ( ? * � " � � � , (matrix product). However,� �� � � � � � � � � � � � ��� � � � � � �� � � ��� � � � � �� 4� � � � � � � � � � � ��

�� 4� � �� � � � � �

��� 4�

� � 4� � � � � � � � � �� � � � � � � � 4 � � � � � � � � � �� �

so� � � � � � � ' � � � ,

� � � � (8.9)

for some phase factor � � ' � � � , . (Note the reversal of order in (8.9).) It fol-lows from (8.9) that the operators � � determine a projective representation of? * � " � � � , i.e., a rep up to a phase factor.

It is easy to verify the operator identity� � � � � 4 � � � ��� � � � � � � � � � � � � � � � � � ��� � � � � (8.10)

where # 4 � � �&�� � �� � �

and� � � � � ����� � � ;� � � % � ����� .

Furthermore, defining the unitary operator� � � � by� � � � � �����&� � 4 8 � � ����� � � ( * � � � � � � � � �

we find � � 4 � � � � � � ��� � � � � � � � � � � � � � � � � � 4 � � � � � � � � ' � , � � � (8.11)

where # � � � �&��� �� �

� 4 � Matrices of the form # 4 � � � , # � � � � generate a two-dimensional subgroup of

? * � " � � � . To generate the full group we need a third one-parameter subgroup.

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We have already derived such operators, the �� 4 � in (4.44). However, from the

form (4.44) it is not easy to verify relations (8.7). It is much easier to use theunitary transformation

� % * � � � � � �to realize the rep � � on the Bargmann-

Segal Hilbert space�

. Recall that �� � 4 � � � �

� 4 � � � 4 takes the form (4.42):

�� � 4 � � � 7 � � � � � 32 7 � � � ( �

The action of � � on�

is given by (4.36):

� � � � � � � 7 � � ��� + � 4< � � � � � � � � � � � � � " � 4 8 � � � ��" � � � � � 7� � � � � � � " � � � � - � � 7 �%" � 4 8 � � � � " � � � � � ��

Now it is easy to verify the identity

���� � 4 � � � � ��� � � � � � � � 4 �� � � + � � � � 4 � " � � �&��� � 4 � � ���� � ��� � 4 � � � � � 4 �

� � 4� � � � � �=4 � " � � �&� � � 4 � � � ���� � � ��� 4 � � � � � 4 � � 4� � � -� � ��� ' � � 2 , � ��� (8.12)

where # � � � � 4 � � � � � � 4 " � � ��� � 4� � ��� 4 #" � � � � � 4 �

Transforming back to* � � � � we see that the operators �

� 4 � in (4.44) must satisfy

�� 4 � � 4 � � � ��� � � 4 ��� � � � � ' � � 2 , � � �

Since�� �� �

� 4 �� � � �=4 � � � 4� � ��� 4 � � � 4 �

��� 4 �� � � �

� � � � 4 � � � ��� 4� �

� � � ��� 4 � � �=4 � �setting �

� � " � � in the case where� � � we find the operator� � 4 � ��� � � " � � � �

� 4 ��� � �� " � � �or� � 4 � � ����� � ( �

" � � �� � � �� � � � �� � � � � �� � � ��� 4 � � 4 8 � ��� � � � � � � ��� 4 �

� � � � � �" � � � � � � �� � � 4 � � � � � � � �(8.13)

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where 4 � " � � 0 ,

an integer, � � � , � �/0 � � and � ( * � � � � . Here� � 4 � satisfies � � �;4 � � � � � � � � 4 � � � � � � ' 2 , � � �

where # � � 4 � � � � � � 4 � ��� 4� � � � 4 � � � 4 �

NOTE: The operators � ��� � � % � � � � � and� � 4 � do not generate a rep of

? * � " � � �but the first two types of operators and the operators

� ��� 4 � ��� � 4 � � � 32 8 � dogenerate a rep of a two-fold covering group

? * � " � � � of

? * � " � � � . This rep iscalled the metaplectic representation. See [[80]] and [[98]] for more details.For " � � � � the rep � � of � � together with the metaplectic rep of

? * � " � � � ex-tends uniquely to an irred unitary rep of the 6-parameter Schrodinger group, thesemi-direct product of ��� and

? * � " � � � . The Schrodinger group is the symmetrygroup of the time-dependent Schrodinger equations for each of the free-particle,the harmonic oscillator and the linear potential in two-dimensional space time.See [[80]] for a detailed analysis.

The formula� � �� �� � �

� � � � ��� � � � ��� � � � �

� � � � � �� � � � � �� � � � � � � � � � �

shows that the unitary operator� � � � corresponding to the matrix # � � � �� � �

can be defined (unique to within a phase factor) by

� � � � � � � � � � � 4��� � � � � ��� � � � � � � � � � � �� � � � � � � � � � �� � ��� � � � � '"% � �7, ��

� � � � � � � �where �

� � ��� � � � . Clearly this operator is well defined and unitary for � � � , � ,since it is a product of unitary operators. Indeed

� � � � is well defined and unitaryfor all real � . To show this we use the fact that the family of all functions ofthe form � � � � � � � � ' � � �7, , for � � �

and � real, spans* � � � � , i.e., the set of all

dilations and translations of � � � spans* � � � � . (See [[62], page 494 ] and Exercise

8.3) Now the integral

�,� � � � � � � � ( �" � � � � � � � � ' � � �+, 8 �� � � � � � � � (8.14)

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is well-defined for all � and agrees with the preceding integral for � � ��, � . Anexplicit evaluation of the integral yields

� � � � � � � ��� �� � � �� ��

� �� � � ' � � � , 8 ':4 � � ���� � , � (8.15)

so � � � � � � 4 � � � � � � � � � �� 4 � � � � � ( �

� 4 � � � ��� � � ' � � � � , 8 ' � � � � ,

(Here the parameters � � � correspond to the functions� .) Furthermore,

� � � 4 � � �,� � � � � � � � �&� � � � �� � '�

� � � ,� � � � 4 8 � ��� � � � � � � � � � � � � �� � '�

� � � ,� � � �� �,� � 4 � � � � � � � � �so

� � � 4 � �,� � � �&� � � � 4 �� � �for all � 4 � � � . Since

� � � � is unitary for � � � , � it follows easily that� � � � is unitary

for all � . Note also that� � � ��� � .

By explicit differentiation in (8.15) we see that for � � ��� � �&� � � � � � � � � , �� � �

��� � � � � � , � � ��� � � � � � � � . Thus,�,� � � � � � � gives the unique solution of the Cauchy

problem for the time dependent free particle Schrodinger equation. In particular

� � � � � � � ��� � � � � � �� � � � � �

where � � � is the unitary operator generated by the self-adjoint operator � via thespectral theorem, [[62]]. In [[80]] it is shown that the Schrodinger group actsas the symmetry group of the time dependent Schrodinger equation, i.e., it mapssolutions into solutions of this equation, and that the possible solutions which areobtainable by separation of variables can be characterized by the group action.Similarly, the operator

� � � � � satisfies the equation

�� � � �" � �

�� � � � � �

� � � � � � ��� � � � � � � �where � � ��� � �&��� ��� � � � � � � . Thus, � is the unique solution of the Cauchy problemfor the time dependent Schrodinger equation equation with a harmonic oscillatorpotential, [[80]]. Such considerations are beyond the scope of these notes.

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8.3 Theta functions and the lattice Hilbert space

For another application of the use of the metaplectic formula (8.6) let us reconsiderour construction of the lattice representation of � � . According to (4.46) this repis defined on functions

� � ��� � � � 9 � � � � on ��� such that

� � � 4 � � 4 � � � � � � � � � � � � � � 4 � � � � � ���� � � � � � 4 � � � � � � � (8.16)

where � 4 � � � are integers. For�

an automorphism (8.6) of � � , with 4 - � 0 � � � ,it is natural to look for the conditions such that � �

� � � 0 � � � � � � � belongs to thelattice Hilbert space for every � belonging to this Hilbert space. If

�corresponds

to the matrix� 4 0� - � � 4 - �10 � � � �

the conditions are

� ) 4 � 4 � 0 � � � 4 � 4 � 0 � � � � � 4 � - � � � � � 4 � - � � �� � � � � � � 4 � � � 4� � 4 � 4 � 0 � � � 4 � 4 � 0 � � �'� � � 4 � - � � � � � 4 � - � � �� 4� � � 4 � � 4 � � � � � � � � /� � ���� � � � � 4 � 4 � 0 � � � � � 4 � - � � � � � � 4� � 4 � 4 � 0 � � � � � � 4 � - � � � � 4� � 4 � � � (8.17)

Since � satisfies only (8.16) we see that 4 � 0 � � � - must be integers. Then (8.16)implies

� � �!4 � 4 � 0 � � � � �&4 � 4 � 0 � � � � � � � 4 � - � � � � � � � 4 � - � � � � )� � � )� �� �&4 � 4 � 0 � � � � � � 4 � - � � � �&� � ���� �� � � � 4 � 4 � 0 � � � � � 4 � - � � � )� � �� (8.18)

Setting )� � � � � � 4� �!4 � 4 � 0 � � � � � � 4 � - � � � � 4� � 4 � � �()� � � � � � 4� � 4 � 4 � 0 � � �'� � � 4 �- � � � � 4� � 4 � � in (8.18), we recover (8.17) provided� 4 � 4 � 0 � � �'� � � 4 � - � � � � � 4 � � �4 � � � 4 � "�0 � � 4 � � � 0 - � �� is an even integer for all integers � 4 � � � . This will be the

case if and only if 4 � 0 0 - 0 � � � �" � (8.19)

i.e., 4 � and 0 - must be even integers.

Thus we see that if # � � 4 0� - � ( ? * � " � � � , i.e., if # ( ? * � " � � � and the

matrix elements of # are integers, and if conditions (8.19) are satisfied, then � ���

117

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belongs to the lattice Hilbert space whenever � so belongs. Reduced to the spaceof functions � � � 4 � � � � where � � � 4 � � � � � � �&� � � � 4 � � � � � ���� ��� , so that

� � � 4 � � 4 � � � � � � � � � � ���& � � � � � � 4 � � � � (8.20)

for � 4 � � � ( � , the action is

����� � 4 � � � �&� � � 4 � 4 � 0 � � � � � 4 � - � � � � �� ) ' 2 � � � 6 � , ' � � � � � � , � � � � * (8.21)

The action of

? * � " � � � on the lattice Hilbert space will lead us to a number ofinteresting transformation formulas for Theta functions.

As we showed earlier, (6.10), the ground state wave function� � ����� �

� � 4 8 < � � � 8 � ( * � � � � is mapped by the Weil-Brezin-Zak transform to

� � � � 4 � � � � � � � 4 8 < � � � � 8 � � � � � � � � 4" � � �" � � (8.22)

where � is the Jacobi theta function, [[40]], [[110]],

� � � � � �&� � � ��� � � � � � � � " � � � � � (8.23)

Here for � such that Im � � � , � is an entire function of � . Moreover, the function�

� ������� � �� � % (+* � � � � , with Im � � � is mapped to

��

� � � 4 � � � �&� � �

� � � 4 � � � �&� � �& � � � � � � 4 � � � � � � � (8.24)

in the lattice Hilbert space. An elementary complex variable argument [[110]]shows that ��

� � � 4 � � � � vanishes precisely once in the square� , � 4 � � , � ,

� � � � , with a simple zero at the point� 4� � 4� � . Thus by Theorem 17 the functions

��

� � � 4 � � � � � ���& (' � � � � � � � , � � 4 � � � ( � , span the lattice Hilbert space. Anotherway to state this is to say that every element in the lattice Hilbert space can bewritten in the form ��

� � � 4 � � � � � � � 4 � � � � where � is a periodic function in � 4 and� � % � � � 4 � � 4 � � � � � � � � � � � 4 � � � � for integers � 4 � � � . Since ��

belongs tothe lattice Hilbert space, so does ��

��� where # ( ? * � " � � � , and satisfies (8.19).Thus ��

���� � �&� ��

� � � � � � � � � (8.25)

for some periodic function ��

. Expression (8.25) describes the framework for afamily of transformation formulas obeyed by the Theta functions. Note that �

�118

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need not be the same as � . In the derivations to follow we will choose ��

so thatthe expressions for �

� are as simple as possible.

As a nontrivial example we take the case # � � � �� � � � , see Exercise 9.3.

The result is

� � ���� � � � � �� � � � � � � � � � 4 � � �&� � � � � �� ( �� � ) � � 4� � � � � � �� /

(We have chosen � � � ��� � . ) This is equivalent to the transformation for-mula

� � ��� � �&� (�� � � �� � 8 � � � �� � � �� ��

(8.26)

As a second example we take # � � � �" � � . Then

��

���� � � � � ���� � � � �� � � � � � � 4 � � " � 4 � � � � � �� � ���� � � � �� � � � � � � ;� � � � ���& � ' � � � � � � � � � ,� � ���� � � � � � ;� � ' � � � � , � ���� � ';� � � � 4 8 � , � � � � ;� ' � � � , ' � � � � , � ���� � � � � ;� � � � � � � � � 4 � � � � � � � � � � � � � "

Thus, � � � � � �&� � � � � � � " �� (8.27)

Even in the cases where the parity conditions (8.19) don’t hold, we get useful

information. For example, consider the case # � � � �� � � . With this automor-

phism we are replacing the function � � � 4 � � � �� � � 4 � � � � � �&� � � � 4 � � � � � � � � 4 � � � � � � � � � ���� � � � � 4 � � � � (8.28)

in the lattice Hilbert space by the function

� � � 4 � � � � � � � � 4 � � 4 � � � � � �& � � Now it is easy to check that

� � � 4 � � � � � � � � � � 4 � � � � � � � � 4 � � � � � �&� � � � ���� � � � � 4 � � � � � (8.29)

so � doesn’t belong to the lattice Hilbert space. However it is straightforwardto show that

)�

� � � 4 � � � � � ��

� �� 4 � � � � 4� � transforms according to (8.29). It

119

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follows from this remark that any square integrable (on the unit square) function� satisfying (8.29) can be written in the form ��

� � � 4 � � � � 4� � �� � � 4 � � � � where

�� is periodic in � 4 � � � . Indeed we find

��

� �� � � � � �� � � � �� � � � � � � 4 � � � 4 � � � � � ��� � �� � � � �� � � � � � � ;�

�� � ���� � ' � � � � � � � � ,� � �� � � � � � 2� � ' � � � � , � ���� � ' � � � � , � � � � ;��' � � 4 , ' � � � � , � ���& � � � � � �� � ;� � � � � � � 4 � � � � � � 4� � � � � � � � � � � �

(Here we have used the fact that � � ;� � � � 2� � for any integer � .) Thus we havethe transformation formula

� � � � � � � � )�� � �" � � � � / (8.30)

Note that by using (8.30) twice we get (8.27), in accordance with the fact that� � �� � � � �

� � �" � � Note: The other three basic Jacobi Theta functions 4 , � , and < (or � ) can easilybe expressed in terms of � , [[40]], [[110]].

Since the modular group elements� � �� � �

and� � ���� � �

generate

? * � " � � � , see for example [[56], pages 168-171], it follows that all the? * � " � � � transformation formulas can be derived by repeated use of (8.26) and(8.30). See [[6]] and [[40]] for details. It is worth remarking that the appropriate� � corresponding to each

� 4 0� - � ( ? * � " � � � is

� � � - � � �0 � � 4 In the preceding discussion we have been concerned with nonorthogonal bases

for the lattice Hilbert space. For completeness we also compute the� �

basis

� � � � 4 � � � �&� � ��� � 4 � � � � � �

� � � � � " � . . .120

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corresponding, via the Weyl-Brezin-Zak transform, to the���

basis (4.27) for* � � � � : �������&� � � 4 8 < � ��� � � 4 8 � � � � � � " � � 8 � � � % 8 � � �

����� �where � �

�����is a Hermite polynomial. We have already seen that the ground state

wave function� � ������� � � 4 8 < � � % 8 � maps to, (8.22),

� � � � 4 � � � �&� � � 4 8 < � � � � 8 � � � � � � � � 4" � � �" � �� Applying the transform � to both sides of the generating function (5.25) for the�������

and using the fact that

� � � � 4 � � � �&� � � 4 8 < � � 6 � ��6 � � � � � � � ) � � � �" � � � 4 � "�0 � � �" � /for � �����&� � � 4 8 < ��� � � 0 � �%" 0 � � 4� � � � , we obtain

� � 4 8 < ��� � � 0 � �%" 0 � 4 � 4� � � 4 � � � � � � ��� � � 4 � "�0 � � ��� �� � � � � � � 6 �' � , ��� � � � � 4 � � � ��

The left-hand side of this expression is an entire function of 0 .With this brief look at the Schrodinger group, an interesting group for future

study which contains both ��� and the affine group as subgroups, we concludethese notes.

8.4 Exercises

1. Compute the automorphism group of � � . Does � � have any outer auto-morphisms?

2. For � � � verify the identity

# � � 4 0� - � � # 4 � 4 � � # � � �" � � � � " � # � � � � # 4 � -� ��

Find a similar factorization for � � � and � � �. Show that the automor-

phisms of � � are generated by # 4 � 4 � , # � � 4��� � � � � � , and # � � � � .121

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3. Apply the automorphism # � � � �� � � � to the Theta function (8.24) in

the lattice Hilbert space to derive the formula

� � ���& � � � � �& � � � � � � � � � 4 � � ��� � � � � �� ( �� � ) � � 4� � � � � � �� /

4. Show that the functions � �& � � � � � � 4 � � � � � � � � ���� (' � � � � � � � , form an���

basis for the lattice Hilbert space. What is the corresponding� �

basis for* � � � � under the inverse Weil-Brezin-Zak transform?

5. Express the relation, Exercise 4.7,

�� " � �

� � �� � �� �

� � �� ��� � %

as a Theta function identity. Compare with equation (8.26).

122

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