Wavelet Regularity 05 236 by M. Pollicott and H. Weiss June 9, 2005

8/9/2019 Wavelet Regularity 05 236 by M. Pollicott and H. Weiss June 9, 2005

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2 M. POLLICOTT AND H. WEISS

window. This is achieved by integer translations and dyadic dilations of a single function:the wavelet.

The special class of orthogonal wavelets are those for which the translations and dilations

of a fixed function, say, form an orthonormal basis ofL2

(R). Examples include Haarwavelet, Shannon wavelet, Meyer wavelet, Battle-Lemari wavelets, Daubechies wavelets,and Coiffman wavelets. A standard way to construct an orthogonal wavelet is to solve thedilation equation

(x) =

2

k=

ck(2x k), (0.1)

with the normalization

kck = 1. Very few solutions of the dilation equation for waveletsare known to have closed form expressions. This is one reason why it is difficult to de-termine the regularity of wavelets.1 Provided the solution satisfies some additionalconditions, the function defined by

(x) =

n=

(1)kc1k(2x k),

is an orthogonal wavelet, i.e., the set{2j/2(2jx n) : j, nZ} is an orthonormal basisfor L2(R). Henceforth, the term wavelet will mean orthogonal wavelet.

Daubechies [Da1] made the breakthrough of constructing smoother compactly supportedwavelets. For these wavelets, the wavelet coefficients {ck} satisfy simple recursion relations,and thus can be quickly computed. However, the Daubechies wavelets necessarily possessonly finite smoothness.2 Smoother wavelets provide sharper frequency resolution of func-

tions. Also, the regularity of the wavelet determines the speed of convergence of the cascadealgorithm [Da1], which is another standard method to construct the wavelet from equation(0.1). Thus knowing the smoothness or regularity of wavelets has important practical andtheoretical consequences.

The main goals of this paper are to extend, refine, and unify the thermodynamic approachto the regularity of wavelets and to devise a faster algorithm for estimating regularity.

In Section 1 we recall the construction of wavelets using multiresolution analysis. Wethen discuss how the Lp-Sobolev regularity of wavelets is related to the thermodynamicpressure of the doubling map E2 : [0, 2)[0, 2). The latter is the crucial link betweenwavelets and dynamical systems.

Previous authors have noted that for compactly supported wavelets, the transfer op-erator preserves a finite dimensional subspace and is thus represented by a d d matrix.Providing the matrix is small, its maximal eigenvalue, which is related to the pressure, canbe explicitly computed, and thus the Lp-Sobolev regularity can be determined by matrixalgebra.

1We learned in [Da2] (see references) that this dilation equation arises in other areas, including sub-division schemes for computer aided design, where the goal is the fast generation of smooth curves andsurface.

2Frequently, increasing the support of the function leads to increased smoothness.


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HOW SMOOTH IS YOUR WAVELET? 3

Cohen and Daubechies identified theLp-Sobolev regularity in terms of the smallest zeroof the Fredholm-Ruelle determinant for the transfer operator acting on a certain Hilbertspace of analytic functions [CD1, Da2]. Their analysis leads to an algorithm for the Sobolev

regularity of wavelets having analytic filters that converges with exponential speed. InSection 2, we study the transfer operator acting on the Bergman space of analytic functionsand effect a more refined analysis. This leads to a more efficient algorithm whichconvergeswith super-exponential speed to the Lp-Sobolev regularity of the wavelet. The algorithminvolves computing certain orbital averages over the periodic points of the doubling mapE2.

In Section 3 we illustrate, in a number of examples, the advantage of this methodover the earlier Cohen-Daubechies method. Running on a fast desktop computer, thisalgorithm provides a highly accurate approximation for the Sobolev regularity in less thantwo minutes.

A feature of this thermodynamic approach is that it works equally well for compactlysupported and non-compactly supported wavelets (e.g., Butterworth filters). In practice,for wavelets with large support, the distinction may become immaterial, and the use ofour algorithm which applies to arbitrary wavelets may prove quite useful.

In Section 4, we use this improved algorithm to study the parameter dependence of theLp-Sobolev regularity in several smooth families of wavelets. In particular, we constructnew examples of wavelets having the same support as Daubechies wavelets, but with higherregularity.

In Section 5, we show that the Lp-Sobolev regularity depends smoothly on the wavelet,and we provide, via thermodynamic formalism, a variational formula. This problem hasbeen considered experimentally by Daubechies [Da1] and Ojanen [Oj1] for different param-

eterized families. Our thermodynamic approach, based on properties of pressure, gives asound theoretical foundation for this analysis.

In Section 6, we study the asymptotic behavior of the Lp-Sobolev regularity of waveletfamilies. We establish a general relation between this asymptotic regularity and maximalmeasures for the doubling map, extending work of Cohen and Conze [CC1]. The study ofmaximal measures is currently an active research area in dynamical systems.

In Section 7, we describe the natural generalization of these results to arbitrary dimen-sions.

Finally, in Section 8, we briefly describe the natural generalization of these results towavelets whose filters are not analytic.

1. Wavelets: construction and regularity

Constructing Wavelets. We begin by recalling the construction of wavelets via mul-tiresolution analysis. A comprehensive reference for wavelets is [Da1], which also containsan extensive bibliography.

The Fourier transform of the scaling equation (0.1) satisfies

() =m(/2)(/2), where m() = 12

k

ckeik . (1.1)


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The function m is sometimes called the filter defined by the scaling equation. Iteratingthis expression, and assuming that m(0) = 1, one obtains that

() = j=1

m(2j). (1.2)

Conditions (1)(3) below allow one to invert this Fourier transform and obtain the scalingfunction. The wavelet is then explicitly given by

(t) =

2

k=

dk(2t k), where dk = (1)k1c1k, (1.3)

and thus the family

j,k = 2j/2(2jt k) , j , k Zforms an orthonormal basis ofL2(R).3

A major result in the theory is the following.

Theorem 1.1 [Da1, p.186]. Consider an analytic 2-periodic function m with Fourierexpansionm() =

n cne

in satisfying the following conditions:

(1) m(0) = 1;(2)|m()|2 + |m(+ )|2 = 1; and(3)|m()|2 has no zeros in the interval [

3,

3].

Then the function defined in(1.2) is the Fourier transform of a functionL2(R) andthe function defined by(1.3) defines a wavelet.

Condition (1) is equivalent to(0)= 0, which is necessary to ensure the completenessin the multiresolution defined by . Condition (1) is also necessary for the infinite product(1.2) to converge. Condition (2) is necessary for the wavelet to be orthogonal, and condition(3), called the Cohen condition, is a sufficient condition for the wavelet to be orthogonal[Da1, Cor. 6.3.2].

If one wishes to construct compactly supported wavelets, then one needs the additionalassumption:

(4) There exists N 1 withcn = 0 for|n| N.Ifck= 0 for k [N1, N2], then the support of is also contained in [N1, N2], and the

support of is contained in [(N1 N2+ 1)/2, (N2 N1+ 1)/2].It is frequently useful to work with the function q() = |m()|2. This function is

positive, analytic, and even. Condition (1) is equivalent to q(0) = 1 and (2) is equivalenttoq() + q(+ ) = 1. A simple exercise shows that (1) is equivalent to

kZ ck = 1/2 and

(2) is equivalent to q() = 1/2 +

kZ c2k+1cos((2k + 1)). Thus the setKof positive even3Engineers often call the family {ck, dk} a quadratic mirror filter (QMF).


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analytic functions satisfying both (1) and (2) is a convex subset of the positive analyticfunctions.

Example (Daubechies wavelet family).

This family of continuous and compactly supported waveletsDaubN {N N}, is obtainedby choosing the filter m such that

|m()|2 = cos2N(/2)PN(sin2(/2)), where PN(y) =N1k=0

N1+k

k

yk.

The wavelets in this family have no known closed form expression.Note that|m()|2 vanishes to order 2N at = . More generally, a trigonometric

polynomial m with|m()|2 = cos2N2

u() satisfies (2) only if u() = Q(sin2(/2)),

where Q(y) = PN(y) +yNR(1/2

y), and R is an odd polynomial, chosen such that

Q(y)0 for 0y1 [Da1, 171].Wavelet regularity. Since the Fourier transform is such a useful tool for constructingwavelets, and since wavelets (and other solutions of the dilation equation) rarely haveclosed form expressions, it seems useful to measure the regularity of wavelets using theFourier transform. For p1, the Lp-Sobolov regularity of is defined by

sp(f) = sup

s: (1 + ||p)s|f()|p L1(R) .The most commonly studied case is p= 2, where ifs2(f)> 1/2, thenf Cs21/2.

One can also easily see that

sp(f) sq(f) 1p

1q

, for 0< p < q.

Thermodynamic Approach to Wavelet Regularity. We now discuss a thermody-namic formalism approach to wavelet regularity. We assume that|m()|2 has a maximalzero of orderM at and write

|m()|2 = cos2M(/2)r(), (1.1)whererC([0, 2)) is an analytic function with no zeros at. We further assume thatr >0, and thus log rC([0, 2)).

Given a functiong : [0, 2)R

and the doubling mapE2 : [0, 2)[0, 2) defined byE2(x) = 2x(mod 2), the pressure ofg is defined by

P(g) = sup

h() +

20

g d : a E2-invariant probability measure

,

where h() is the measure theoretic entropy of . If g is Holder continuous, then thissupremum is realized by a unique E2invariant measure g, called the equilibrium statefor g.

The following theorem expresses the relationship between the pressure and the Sobolevregularitysp(), and provides the crucial link between wavelets and dynamical systems.


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Theorem 1.2. TheLp-Sobolov regularity satisfies the expression

sp() =M

P(p log r)

p log2

.

Proof. The proof for p = 2 is due to Cohen and Daubechies [CD1] and the general caseis due to Eirola [Ei1] and Villemoes [Vi1]. See also [He1, He2]. This result was originallyformulated in terms of the spectra radius of the transfer operator on continuous functions;but this is precisely the pressure (see Proposition 2.1).

See Appendix I for a proof of in arbitrary dimensions.

2. Determinants and transfer operators

Given h: [0, 2) Rwe define the semi-norm

|h| = supx=y

|h(x) h(y)||x y| ,

and the Holder norm||h|| =|h|+ |h|, where 0<


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The Ruelle-Fredholm determinant for Lg can be expressed as

det(I zLg) = expn=1

zn

n En2x=x

exp(Sng(x))

1 2n , (2.1)and has the usual determinant property det(I zLg) =

n=1(1 zn), where n

denotes the n-th eigenvalue ofLg. Note that this determinant is formally quite similar tothe Ruelle zeta function, and has an additional factor of the form 1 /(1 2n) in the innersum. More generally, Lg need not be trace class for expression (2.1) to be well defined forsmall z (see Appendix III).

For our applications to wavelet regularity, we consider g =p log r (see (1.1)). To com-pute the determinant, one needs to compute the orbital averages of the function g over all

the periodic points ofE2. Every root of unity of order 2

n

1 is a periodic point for E2,and there are exactly 2n 1 such roots of unity, hence there are exactly 2n 1 periodicpoints of period n for E2. A routine calculation yields

dp(z) := det(I zLplogr) = 1 +n=1

bnzn, where

bn=

n1+...+nk=n

(1)k apn1 apnk

n1! nk! and apn =

1

1 2n 2n1

k=0

n1j=0

r

2jk

2n 1

p

.

The next theorem summarizes some important properties of this determinant dp(z).

Theorem 2.2. Assume log rC[0, 2). Then(1) The functiondp(z) can be extended to an entire function ofz and a real-analytic

function ofp;(2) The reciprocal of the exponential of pressurezp:= exp(P(p log r))) is the smallest

zero fordp(z);(3) The Taylor coefficients bn of dp(z) decay to zero at asuper-exponential rate,

i.e., there exists0< A


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where n denotes the n-th eigenvalue ofLp log r [Sim]. It follows that the coefficient bn =i1


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N Smallest root of d2 Approx to s2(Daub3 )

1 0.50000000000000000 2.50000000000000002 0.11590073269851241 1.4454807959406297

3 0.10935186260699924 1.40352484967659804 0.11120565746767627 1.41565104529629385 0.11111111111111109 1.41503749927884376 0.11111111111111111 1.4150374992788437

Example 2: (N= 2andp = 4). In this cased4(z) is a polynomial of degree 5 with rationalcofficients:

d4(z) = 1 2z 50.75z2 76.25z3 + 115z4 32z5,which has its first zero atz0 = 0.115146 . . . . By Theorem 2.1 we deduce that s4(

Daub2 ) =

3 + log(0.115146 . . . )/2 log 2 = 1.22038 . . . .6

Example 3: (N= 4andp = 2). In this cased2(z) is a polynomial of degree 7 with rationalcoefficients

d2(z) = 1 2z 433z2 996z3 + 23572z4 32288z5 68627z6 24414z7,

which has its first zero atz0 = 0.045788 . . . . By Theorem 2.1 we deduce that s2(Daub4 ) =

4 + log(0.045788 . . . )/2 log 2 = 1.77557 . . . .7

Based on the same approximation scheme, the following table contains our approxima-tions to s2(daubN ) for N = 1, 2, . . . , 10, and compares them with the values obtained byCohen and Daubechies in [CD1].

N new approx to s2(Daub4 ) previous approx to s2(

Daub4 )

1 0.500000 0.3388562 1.000000 0.9998203 1.415037 1.4149474 1.775565 1.7753055 2.096786 2.096541

6 2.388374 2.3880607 2.658660 2.6585698 2.914722 2.9145569 3.161667 3.161380

10 3.402723 3.402546

6This compares with the estimate of Cohen-Daubechies of 1.220150 . . . , which is accurate to threedecimal places.

7This compares with the estimate of Cohen-Daubechies of 1.775305, which is accurate to three decimalplaces.


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Remark. In examples 1 3 the determinant is a rational function. This is an immediateconsequence of the fact that the transfer operator preserves the finite dimensional subspaceof functions spanned by

{1, cos(), cos(2), , cos(N )}.

This fact has been observed by several authors, c.f., [Oj1].

3.2 Butterworth filter family.

The Butterworth filter is the filter type with flattest pass band and which allows amoderate group delay [OS1]. These filters form an important family of non-compactlysupported wavelets and are defined by

|m()|2

= cos2N(/2)

sin2N(/2) + cos2N(/2) , N N.

We suppress the indexnfor notational convenience. It is easy to check that |m()|2 satisfiesconditions (1) (3) on page 4.

Example4: (Butterworth filter with N=2, p=2. ) For this case the determinant d2(z) is apower series of the form

d2(z) = 1

2. z

2.08 z2

0.479053 z3

0.0124521 z4

0.000086246 z5

2.06875 109 z6

3.2482 1013 z7 7.38964 1013 z8 + 4.24431 1012 z9 2.94676 1011 z10 + ,

which has its first zero at z0 = 0.356702 . . . , and thus by Theorem 1.2 we deduce thats2(

Butt2 ) = 2 + log(0.356702 . . . )/2 log 2 = 1.25497 . . . .

The following table contains the first few approximations ofs2(Butt2 ):

N Root Approx to s2(Butt2 )

1 0.500000000000000 1.00000000000000002 1.256072268804003 1.26897570550919533 0.356757200613385 1.25650722688040034 0.356702225093496 1.25639606020824735 0.356702089349239 1.25639578569691866 0.356702089348077 1.25639578569456987 0.356702089348077 1.2563957856945698

We can similarly compute s2(ButtN ) for N = 1, 2, . . . , 10 and compare these with the

values in [CD1].


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N new approx to s2(ButtN ) previous approx to s2(

ButtN )

1 0.500000 0.3388562 1.256211 1.256395

3 2.044109 2.0441174 2.843768 2.8437715 3.648646 3.6488176 4.456118 4.4562527 5.264533 5.2645868 6.072947 6.0730349 6.881125 6.881173

10 7.688598 7.688776

4. Compactly supported wavelets smoother than the Daubechies family

We now consider smooth finite dimensional families of wavelets and attempt to findthe maximally regular wavelets in the family. Several authors have used this approach tofind compactly supported wavelets that are more regular than the Daubechies waveletswith the same support. The most straightforward approach to the problem of maximizingthe regularity of compactly supported wavelets in parameterized families reduces to theproblem of maximizing the maximal eigenvalue of an N N matrix. However, exceptfor very small N, there is no algebraic expression for the eigenvalue. In this section, weillustrate the advantages of our fast converging algorithm in seeking more regular wavelets.

Example 1. Ojanen [Oj1] experimentally studied the regularity of wavelets corresponding

to filters |m()|2

of widthNand which vanish to order Mat. In the case that M N1he found that the Daubechies polynomials (where M=N) are apparently the smoothest.ForN= 2 and M= 1, say, one can define the family of filters

|m()|2 = 12

(1 + cos ) (1 + 8a cos2 8a cos ) :=r()

. (4.1)

The transfer operator preserves the subspace spanned by{1, cos(), cos(2)}and thus canbe represented by a 3 3 matrix whose maximal eigenvalue is exp P = 1 + 1 + 16a. Inthe special case when a=1/16 one obtains the filter corresponding to Daub2 , for whichr() = 0. One can easily show that this wavelet is the smoothest in this family. However,for the broader classM N 2, this is no longer the case.

Example 2. We consider the two-parameter family of putative waveletscorresponding tofilters

|m(x)|2 =12

+ a cos(x) + b cos(3x) +

1

2 a b

cos(5x). (4.2)

We say putative, because all the conditions in Theorem 1.1 need to be verified for allparameter values. The Daubechies waveletDaub3 corresponds to parametersa= 75/128andb=25/256.


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Here|m(x)|2 has a double zero at x= and thus one can always take out (at least) afactor of (1 + cos()) /2. However, it is always necessary to check by hand that for eachparticular choice ofaandb expression (4.2) is positive and the Cohen condition holds.

It is useful to write (4.2) in the form r()(1 + cos())/2, where

r() = 1 4 (1 + 2 a + 4 b) cos() + 4 (1 + 2 a + 4 b) cos2()+ 16 (1 + 2 a + 2 b) cos3() 16 (1 + 2 a + 2 b) cos4().

Notice this trigonometric polynomial has the form r() = 1 cos() + cos2() +cos3() cos4().

(A) The transfer matrix approachWe consider the finite dimensional space spanned by{1, cos(), cos2(), cos3(), cos4()}

and write the action of the transfer operator associated to weights of the general form

r() = 1 cos() + cos2() + cos3() cos4().

A routine calculation yields that the transfer operator on this space has the followingmatrix representation:

Llog r =Llog r(, ) =

2 + 2 2 0 0 + 2 + 2 0 0

1 + 2 4 1 + 34 2 34 4 0

2

+ 4

+ 3

4

2

+ 34

4

0

12 + 4 8 1 + 34 2 12 + 34 34 4 2 8

.

The matrix corresponding to Daub3 is

Llog r(7/8, 3/8) =

2116 12 316 0 01116

12

316 0 0

2132

1332

532

332 0

1132

1932

532

332

02164

1732

18

132

364

,

with eigenvalues 1, 9/16,

1/4 and 3/64. Then det(I

zLlog r), has a zero at z = 16/9.By Theorem 1.2 we again recover the Sobolev regularity of Daub3 : s2(

Daub3 ) = 1

(log(9/16)/(2 log 2) = 1.41504 . . . .By varying the parametersand, and always checking that expression (4.2) is positive

and satisfies the Cohen condition, one can search for wavelets having maximal LpSobolevregularity, and, in particular, having greater regularity than the Daubechies wavelet. Thetwo required conditions make this a difficult constrained optimization problem, which iseven difficult numerically. In contrast with Example 1, there appears to be no simplealgebraic expression for the eigenvalues for the matrix corresponding to Llog r; one mustresort to empirical searches using different choices of and.


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However, it is more illuminating to implement this computation using determinants.

(B) The determinant approach

If one computes the determinant det(1 zLlog |m|2Daub) associated with the DaubechieswaveletDaub3 using our determinant algorithm, one obtains the power series expansion

det(1 zLlog |m|2Daub) = 1 2 z+ 1.31904 z2 0.35310 z3 + 0.03319 z4

+ 0.00120 z5 0.000351 z6 + .

We observe that:

(1) There are trivial zeros for det(1 zLlog |m|2Daub) at the values z = 1, 2, 4, 8;(2) There is a zero atz = 7.07337 . . . .,which by Proposition 2.3 corresponds to a zero

of det(1

zLlog rDaub) at 7.07337/2

6 = 0.111 . . . . This agrees with our previous

calculation in Section 3.

By Theorem 1.2, if we change the filter|m|2 to decrease the pressure, then the Sobolevregularity of the associated wavelet increases.

One can easily check that if one replaces a = 75/128 by 75/1280.00068, say, andb=25/256 by 25/256+0.00002, say, then the associated filter |m()2| remains positive,and by Theorems 1.2 and 2.1, the zero of the determinant increases from 7.1111 to 8.51989.This reflects an increase in the L2-Sobolev regularity from 1.415 to 1.545.

Thus, this two-parameter family contains wavelets having the same support as Daub3with greater L2-Sobolev regularity. The same idea can be applied for other N.

Remarks: Other approaches.

(a) Daubechies [Da1, p.242] considered the one-parameter family of filters

|m()|2 = cos2(N1)(/2)(PN1(sin2(/2)) + a cos2N(/2)) cos(),

where a R. For definiteness, we discuss the case N = 3. For different values of theparametera we calculate the first zero of the determinant, and thus the Sobolev regularityof the associated wavelet using Theorem 2.2. The following table shows that this zero ismonotone increasing as a increases to 3. The zero of the determinant is 0.40344 . . . whena= 3, which corresponds to L2 Sobolev regularity 1.34443 . . . . It is not necessary to study

the case a > 3 since the expansion for|m|2 is no longer positive. To see this, we canexpand in terms of as

|m()|2 = cos4(/2)(P2(sin2(/2)) + a cos4(/2)) cos()= cos4(/2 /2)(1 + 2 sin2(/2 /2) + a sin6(/2 /2) cos( ))= sin4(/2)(1 + 2 cos2(/2) a cos6(/2)cos())= 1 + 2(1 (/2)2 + . . . )2 a(1 (/2)2 + . . . )6(1 2 + . . . )= (3 a) (1 + 5a/2)2 + . . . .


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It is clear this expression is negative values for a >3.We see that the parameter valuea= 3 corresponding to the maximally regular wavelet

lies on the boundary of an allowable parameter interval, and thus could not be detected

using variational methods.

a Zero of determinant Sobolev regularity

-1.0 0.230 0.939-0.5 0.239 0.9670.0 0.250 1.0000.5 0.262 1.0301.0 0.276 1.0711.5 0.294 1.1162.0 0.317 1.7102.5 0.350 1.2422.7 0.367 1.2762.8 0.377 1.2962.9 0.389 1.3183.0 0.403 1.344

(b) Daubechies [Da3] also considered the one-parameter family of functions

m() =

1 + ei

2

2Q(),

where|

Q()|

2 =P(cos ) andP(x) = 2

x + a

4

(1

x)2, with the parametera ichosen such

that P(x) >0 for 0x


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Proposition 5.1.

(1) For every p 1, the regularity mappingK R defined by|m()|2 sp() isanalytic;

(2) Let |m()|2 = cos2M(/2) (r0() + r0() + . . . )be a smooth parametrized family of filters. Then the change in the Sobolev regularityis given by

sp() =sp(

0) +

1

p log2

r0

r0d

+ . . . ,

where is the unique equilibrium state forp log r0.

Proof. The analyticity of the Sobolev regularity follows from Proposition 2.1(2), the analyt-

icity of the pressure, and the implicit function theorem. Since log r = log(r0+r0 +. . . ) =log r0 + r0r0 + . . . andDgP() =

dg, this follows from the chain rule. If we restrict to finite dimensional spaces we can consider formulae for the second (and

higher) derivatives of the pressure, which translate into formulae for the next term in theexpansion ofsp().

Also observe that we can estimate the derivatives of the Sobolev exponent of a familysp() using the series for dp(, z). More precisely, by the implicit function theorem

dsp()

d =

d(det(, z))

d d(det(, z))

dz =

d(det(, z))

d d(det(, z))

dz + O(2n

2

)

which is possible to compute for periodic orbit data. We can similarly compute the higherderivatives

dksp()

dk

=0

, for k1

using Mathematica, say. We can then expand the power series for the Sobolev function:

sp() =sp(

0) + ( 0) dsp()

d

=0

+ ( 0)2

2

d2sp()

d2

=0

+ . . . .

6. Asymptotic Sobolev regularity

The use of the variational principle to characterize pressure (and thus the Sobolevregularity) leads one naturally to the idea of maximizing measures. More precisely, onecan associate to g:

Q(g) = sup

20

g d, an E2-invariant Borel probability measure

Unlike pressure, which has analytic dependence on the function, the quantity Q(g) is lessregular, but we have the following trivial bound.


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Lemma 6.1. For any continuous functiong: [0, 2)R,

Q(g)P(g)Q(g) + log 2.

Proof. Since 0h() log 2, this follows from the definitions. Proposition 6.2. Letn, nN be a family of wavelets with filters

|m()|2 = cos2Mn(/2)rn(),

whereMn0. Then the asymptoticLp-Sobolev regularity of the family is given by

limn

sp(n)

n = lim

n

Mnn

limn

Q(log rn)

n log2 .

Proof. Using Theorem 1.2, Lemma 6.1, and (1.1), one can easily show

Q(p log rn)p log 2(Mn sp(n))Q(p log rn) + log 2.

The proposition immediately follows after dividing bynand taking the limit asn .

We now apply Proposition 6.2 to compute the Lp-Sobolev regularity of the Daubechiesfamily of wavelets for arbitrary p2. Cohen and Conze [CC1] first proved this result forp= 2.

Proposition 6.3. Let p 2. The asymptotic Lp-Sobolev regularity of the Daubechiesfamily is given by

limN

sp(DaubN )

N = 1 log3

2log2.

Proof. If one applies Proposition 6.2 to the Daubechies filter coefficients (see Section I),one immediately obtains

limN

sp(DaubN )

N = 1

limN

Q(log PN(sin2 (y2 )))

Nlog 2

.

Cohen and Conze [CC1, p. 362] show that

Q(log2 PN(sin2 (

y

2))) =

log PN(3/4)

2log2 ,

and the sup is attained for the period two periodic orbit{1/3, 2/3}. Thus

limN

sp(DaubN )

N = 1 1

2log2 limN

log PN(3/4)

N .


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The exponentially dominant term in PN(3/4) is easily seen to be

2N1N1

( 34 )

N1, and aroutine application of Stirlings formula yields

limN

log PN(3/4)N

= log 3.

The proposition immediately follows.

Remark. As briefly discussed in Section 3.2, the Butterworth filter family ButtN , N Nisthe family of non-compactly supported wavelets defined by the filter

|m()|2 = cos2N(/2)

sin2N(/2) + cos2N(/2).

Fan and Sun [FS1] computed the asymptotic regularity for this wavelet family; theyshowed that

limN+

sp(ButtN )

N =

log3

log2.

This asymptotic linear growth of regularity is similar to that for the Daubechies waveletfamily.

7. Higher dimensions

The dilation equation (0.1) has a natural generalization to d-dimensions of the form:

(x) = det(D)kZd

ck(Dx k), (7.1)

whereDGL(d,Z) is a d dmatrix with all eigenvalues having modulus strictly greaterthan 1. We are interested in solutions L2(Rd). We consider the expanding mapT : [0, 2)d [0, 2)d byT(x) =Dx (mod 1). By taking d-dimensional Fourier transfor-mations, (7.1) gives rise to the equation

() =m(D1)(D1), where m() = kZd ckeik,.Thus by iterating this identity we can write() =n=1 m(Dn).

ForgC([0, 2)d), one can define a transfer operatorLg :C([0, 2)d)C([0, 2)d)by

Lgk(x) =Ty=x

exp

g(y)

k(y),

where the sum is over the det(D) preimages ofx.


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We can consider a Banach space of periodic analytic functionsA in a uniform neigh-bourhood of the torus [0, )d and the associated Bergman space of functions. We havethat for g A the operator Lg :A Aand its iterates Lkg are trace class operators foreach k1. We can compute

tr(Lkg) =

Tnx=x

|det(D)|n exp(Sng(x))|det(Dn I)| ,

whereSng(x) =g(x) + g(T x) + + g(Tn1x).Assume that we can write m(x) =

a(Dx)|det(D)|a(x)

Mr(x), where r(x)>0.8 The Ruelle-

Fredholm determinant ofLp log r takes the form:

det(I Lp log r) = exp n=1

znn

Tnx=x

|det(D)|n exp(pSnlog r(x))|det(Dn I)|

.The statement of Theorem 1.2 has a partial generalization to d-dimensions. We define

the Lp-Sobolev regularity ofp to be

sp() := sup{s >0 :|()|p(1 + ||||p)s L1(Rd)}.Proposition 7.1. TheLp-Sobolev regularity satisfies the inequality

sp() Mdet(D)log

P(p log r)p log

,

where >1 is smallest modulus of an eigenvalue ofD.

Proof. We can write|()| C|det(D)|Mnni=0 r(Di), for some C > 0 and all Dn([0, 2)d). However, we can identify

TnD

ni=0 r(D

i)pd=D

Lnp log r1()d. More-

over, since Lp log r has maximal eigenvalue eP(p log r), we can bound this by CenP(p log r),

for some C >0. In particular,

Rd

|()|p(1 + ||||p)sd n=1

psn TnDTn1D

|()|pdC C

n=1

psn|det(D)|MpnenP(p log r)

In particular, the right hand side converges ifps|det(D)|MpeP(p log r)


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In certain cases, for example ifD has eigenvalues of the same modulus and some othertechnical assumptions, then this inequality becomes an equality [CD1].

As in Section 2, a routine calculation gives that

dp(z) := det(I zLp log r) = 1 +n=1

bnzn, where

bn =

n1++nk

(1)k apn1 apn1

n1! nk! and apn =

Tnx=x

|det(D)|nTnx=x(ni=0 r(x))p|det(Dn I)| .

The simplest case is that of diagonal matrices, but this reduces easily to tensor products(x1, . . . , xd) = 1(x1) d(xd) of the one dimensional case. The next easiest exampleillustrates the non-separable case:

Example. Ford= 2 we let D= 1 1

1 1

, which has eigenvalues =2.

In the d-dimensional setting we have the following version of Theorem 2.2:

Theorem 7.2. Assume that log r A. Then(1) The functiondp(z) can be extended to an entire function ofz and a real analytic

function ofp;(2) The exponential of pressurezp := exp(P(p log r)) is the smallest zero fordp(z);(3) The Taylor coefficientsbn ofdp(z) decay to zero at a super-exponential rate, i.e.,

there exists0< A 0.


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This follows from a standard analysis of transfer operators, and in particular, studyingtheir essential spectral radii. The radius of convergence ofdp(z) is the reciprocal of theessential spectral radius of the operators. In the case that 0 < 1 this requires a slightlydifferent analysis [Ta1]. The necessary results are summerized in the following proposition.

Proposition 8.2.

(1) Assume cn = O(|n|). The transfer operator Lf has essential spectral radius(1/2)eP(f) (and the functiond(z) in analytic in a disk|z|< 2).

(2) Assume cn = O(|n|k). The transfer operator Lf has essential spectral radius(1/2)keP(f) (and the functiond(z) in analytic in a disk|z|< 2k).

The first part is proved in [Po1], [PP2]. The second part is proved in [Ta1], [Ru2] 9

As in Section 2, the speed of decay of the coefficients bn immediately translate into thecorresponding speed of convergence for the main algorithm.

Appendix I: Proof of Theorem 2.1.

Given a bounded linear operatorL : HHon a Hilbert spaceH, itsith approximationnumber(or singular value)10 si(L) is defined as

si(L) = inf{||L K||: rank(K)i 1},

whereK is a bounded linear operator on H.

Let r C denote the open disk of radius r centered at the origin in the complexplane. The phase space [0, 2) for T = E2 is contained in 2+ for any > 0, and thetwo inverse branches T1(x) =

12 x and T2(x) =

12 x+

12 have analytic extensions to 2+

satisfying T1(2+)T2(2+) 2+ 2 . Thus T1 and T2 are strict contractions of2+ onto 2+ 2 with contraction ratio = (2 + )/(2 + 2)< 1. We can choose >0arbitrarily large (and thus arbitrarily close to 1/2).

Let A2(r) denote the Bergman Hilbert space of analytic functions on r with inner

productf, g:=

f(z) g(z) dxdy [Ha1].

Lemma A.1. The approximation numbers of the transfer operator Lp log r : A2(2)A2(2) satisfy

sj(Lp log r)||Lp log r||A2(2)1

j ,

for allj1. This implies that the operatorLp log r is of trace class.Proof. Let g A2(2+) and write Lp log rg =

k= lk(g)pk, where pk(z) = z

k. We

can easily check that||pk||A2(2) =

k+1 (2)

k+1 and||pk||A2(2+) =

k+1 (2 +

9At least when N.10We use these terms interchangably because one can show that sk(L) =

pk(LL).


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for some B >0.

We finish by showing that Theorem 2.1 follows from Lemma 2.2. The coefficients of

det(I zLs) = 1 +n=1 bnzn are given by Cauchys Theorem:|bn| 1

rn sup|z|=r

|det(I zLs)|, for any r >0.

Using Weyls inequality we can deduce that if|z|= r then

|det(I zLs)| j=1

(1 + |z|j)j=1

(1 + |z|sj)

1 + B

m=1(r)mm(m+1)

2 where=||Ls||A2(2). If we choose r=r(n) appropriately then we can get the boundsgiven in the theorem.

Appendix I I: Proof of Proposition 2.3

The periodic orbits of period n for E2 are precisely the sets{2 2ik/(2n 1) : i =0, 1, . . . , n 1}, wherek= 0, . . . , 2n 1. We can now consider the factorization (1.1) andwrite

Zn := 1

1 2n

Tnx=x

n1i=0

|m(2ix)|2

= 1

1 2n2n1k=0

n1i=0

m2 2ik2n 12

= 1

1 2n2n1k=0

n1i=0

cosM

2 2ik

2(2n 1)n1

i=0

r

2

2ik

2(2n 1)

.

However, since cos = 12sin(2)/ sin , we can write

n1i=0

cosM

2 2ik

2(2n 1)

=n1i=0

cosM

2 2ik

2(2n 1)

=

2Mn ifk= 01 ifk = 0

.

Thus we can write

Zn = 1

1 2n

2n1k=0

1

2nM

n1i=0

r

2

2ik

2(2n 1)

+ 1 12nM

.


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Since (1 2nM)/(1 2n) =M1j=0 2jn , we can writedet(1 zLp log |m|2) = exp

n=0

M1j=0

(z2j)ndet1 z2MLp log r=

M1j=0

1 z2j det1 z

2MLp log r

.

References

[CC1] A. Cohen and J.-P. Conze, Regularite des bases dondelettes et mesures ergodiques., Rev. Mat.

Iberoamericana 8 (1992), 351365.[CD1] A. Cohen and I. Daubechies, A new technique to estimate the regularity of refinable functions,

Rev. Mat. Iberoamericana 12 (1996), 527591.[Da1] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied

Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), 1992.

[Da2] I. Daubechies,Using Fredholm determinants to estimate the smoothness of refinable functions ,Approximation theory VIII, Vol. 2 (College Station, TX, 1995), Ser. Approx. Decompos. 6(1995), World Sci. Publishing, 89112.

[Da3] I. Daubechies, Orthonormal Bases of Compactly Supported Wavelets II, SIAM J. Math. Anal.24 (1993), 499519.

[Ei1] T. Eirola, Sobolev characterization of solutions of dilation equations., SIAM J. Math. Anal.23 (1992), 1015103.

[FS1] A. Fan and Q. Sun, Regularity of Butterworth Refinable Functions, preprint.[GGK1] I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of linear operators, vol 1, 1990.

[Ha1] P. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, reprint of thesecond (1957) edition, AMS Chelsea Publishing, Providence, RI,, 1998.

[He1] L. Herve, Comportement asymptotique dans lalgorithme de transformee en ondelettes, Rev.Mat. Iberoamericana 11 (1985), 431451.

[He2] L. Herve, Construction et regularite des fonctions dchel le, SIAM J. Math. Anal. 26 (1995),13611385.

[JP1] O. Jenkinson and M. Pollicott, Computing invariant densities and metric entropy, Comm.Math. Phys. 211 (2000), 687703.

[JP2] O. Jenkinson and M. Pollicott, Orthonormal expansions of invariant densities for expandingmaps, Advances in Mathematics, 192 (2005), 1-34.

[Oj1] H. Ojanen,Orthonormal compactly supported wavelets with optimal sobolev regularity, Applied

and Computational Harmonic Analysis 10 (2001), 93-98.[OS1] A. Oppenheim and R. Shaffer, Digital Signal Processing, Pearson Higher Education, 1986.[PP1] W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structures of Hyperbolic

Dynamics, Asterisque 187188 (1990).

[Po1] M. Pollicott, Meromorphic extensions for generalized zeta functions, Invent. Math 85 (1986),147164.

[Ru1] D. Ruelle, Thermodynamic Formalism, Addison-Wesley, 1978.[Ru2] D. Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes tudes Sci. Publ.

Math. 72 (1990), 175193.

[Ru2] D. Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes tudes Sci. Publ.Math. 72 (1990), 175193.


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[Si1] B. Simon, Trace ideals and their applications, LMS Lecture Note Series, vol. 35, CambridgeUniversity Press, 1979.

[Ta1] F. Tangerman, Meromorphic continuation of Ruelle zeta functions, Ph.D. Thesis, BostonUniversity (1988).

[Vi1] L. Villemo es, Energy moments in time and frequency for two-scale difference equation solu-tions and wavelets, SIAM J. Math. Anal. 23 (1992), 15191543.

[W] Matlab, Wavelet Toolbox Documentation, www.mathworks.com/access/helpdesk/help/tool-box/wavelet/wavelet.html?BB=1.

Wavelet Regularity 05 236 by M. Pollicott and H. Weiss June 9, 2005

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Transcript of Wavelet Regularity 05 236 by M. Pollicott and H. Weiss June 9, 2005