Reproducing Kernels in Hilbert Spaces - math.unl.edujorr1/classes/2009summer/math896/... ·...

ReproducingKernels in Hilbert

Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2

Properties of Kernels

Conclusion

Reproducing Kernels in Hilbert Spaces

Thomas Clark

August 10 - 11, 2009


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Introduction

Definitions

ExamplesR2

`2

H2 and H2

A2

KernelsKernel of `2

Kernel of H2

Kernel of A2


Conclusion


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Resources and Thanks

I This talk is based on the paper Theory of ReproducingKernels by N. Aronszajn. The presenter also referencedA Hilbert Space Problem Book by P. Halmos andFunctional Analysis by J. Conway.

I Special thanks are given to Dr. Orr for recommendingthe paper and helping with the examples.

I Also thanks to Jason Hardin and Zach Roth for TeXsupport.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Some Definitions you may know.

I Suppose F is a class of functions defined on a set E andadmits scaler multiplication by complex(or real)constants. Suppose further that for f ∈ F is defined anorm ‖f ‖, that is, a real number satisfying:

I ‖f ‖ ≥ 0,I ‖f ‖ = 0 iff f = 0,I ‖cf ‖ = ‖c‖ ‖f ‖I ‖f + g‖ ≤ ‖f ‖+ ‖g‖.

I A linear functional is a linear map from a vector spaceto its field of scalers(in our case C or R). We will use Fto denote the field of scalers.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



I ‖f ‖ ≥ 0,

I ‖f ‖ = 0 iff f = 0,I ‖cf ‖ = ‖c‖ ‖f ‖I ‖f + g‖ ≤ ‖f ‖+ ‖g‖.



Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



I ‖f ‖ ≥ 0,I ‖f ‖ = 0 iff f = 0,

I ‖cf ‖ = ‖c‖ ‖f ‖I ‖f + g‖ ≤ ‖f ‖+ ‖g‖.



Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



I ‖f ‖ ≥ 0,I ‖f ‖ = 0 iff f = 0,I ‖cf ‖ = ‖c‖ ‖f ‖

I ‖f + g‖ ≤ ‖f ‖+ ‖g‖.I A linear functional is a linear map from a vector space

to its field of scalers(in our case C or R). We will use Fto denote the field of scalers.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



I ‖f ‖ ≥ 0,I ‖f ‖ = 0 iff f = 0,I ‖cf ‖ = ‖c‖ ‖f ‖I ‖f + g‖ ≤ ‖f ‖+ ‖g‖.



Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

more definitions...

Given a vector space V an inner product on V is a function〈., .〉 : V × V → F s.t. for all α, β in F, and x , y , z in V , thefollowing are satisfied.

I Conjugate Symmetry: 〈x , y〉 = 〈y , x〉.I Linearity of first argument: 〈αx , y〉 = α〈x , y〉 and〈x + y , z〉 = 〈x , z〉+ 〈y , z〉.

I Positive definite: 〈x , x〉 > 0 for all x 6= 0.

I Note that the first two imply conjugate linearity of thesecond argument, i.e. 〈x , αy + βz〉 = α〈x , y〉+ β〈x , z〉


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

more definitions...


I Conjugate Symmetry: 〈x , y〉 = 〈y , x〉.

I Linearity of first argument: 〈αx , y〉 = α〈x , y〉 and〈x + y , z〉 = 〈x , z〉+ 〈y , z〉.




Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

more definitions...


I Conjugate Symmetry: 〈x , y〉 = 〈y , x〉.I Linearity of first argument: 〈αx , y〉 = α〈x , y〉 and〈x + y , z〉 = 〈x , z〉+ 〈y , z〉.




Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

more definitions...

I Note that given an inner product 〈., .〉, a norm isinduced such that ‖x‖ := 〈x , x〉1/2.

I Also any inner product space induces a metric whered(x , y) := ‖x − y‖ so our vector space is a metricspace.

I Thus the class F with the norm ‖ ‖ forms a normedcomplex vector space. If this space is complete withrespect to the induced metric it is a Hilbert Space.

I Note, in general a Hilbert space is assumed to becomplex. There are real Hilbert spaces but they aregenerally given the adjective to note that not allproperties of Hilbert spaces apply to them.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

more definitions...

I A basis for a vector space V is a maximal linearlyindependent set.

I Two vectors x , y ∈ V are orthogonal if 〈x , y〉 = 0.This is written x ⊥ y .

I An orthonormal basis for V is a set E having theproperties

I For each e ∈ E , ‖e‖ = 1.I If e1, e2 ∈ E and e1 6= e2 then e1 ⊥ e2.I E spans V when infinite linear combinations are allowed.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

more definitions...




I For each e ∈ E , ‖e‖ = 1.

I If e1, e2 ∈ E and e1 6= e2 then e1 ⊥ e2.I E spans V when infinite linear combinations are allowed.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

more definitions...




I For each e ∈ E , ‖e‖ = 1.I If e1, e2 ∈ E and e1 6= e2 then e1 ⊥ e2.

I E spans V when infinite linear combinations are allowed.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

more definitions...




I For each e ∈ E , ‖e‖ = 1.I If e1, e2 ∈ E and e1 6= e2 then e1 ⊥ e2.I E spans V when infinite linear combinations are allowed.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Our Main Topic

Let F be a class of functions defined on E , forming a HilbertSpace(complex or real). The function K (x , y) of x and y inE is called a reproducing kernel (r.k.) of F if:

I For every y , K (x , y) as a function of x belongs to F .

I The reproducing property : for every y ∈ E and everyf ∈ F ,

f (y) = 〈f (x),K (x , y)〉xThe subscript x indicates the inner product applies tofunctions of x .


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Good ole R2

Let R2 have the standard dot product,〈(x1, y1), (x2, y2)〉 = x1x2 + y1y2. Then R2 is an innerproduct space, with respect to the induced norm it iscomplete, and thus it is a real Hilbert Space.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Good ole R2


An orthonormal basis for R2 would be {(0, 1), (1, 0)}.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Good ole R2


An orthonormal basis for R2 would be {(0, 1), (1, 0)}.

We should note that R2 is not a complex Hilbert space. Alsowe will mainly be considering Hilbert spaces whose elementsare functions, and so R2 gives us some sense of the structureof a Hilbert space it is not an example of what we will bestudying here.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Little `2

Let `2 = {(xk) : xi ∈ F} such that∞∑

k=0

|xk |2 <∞.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Little `2


k=0

|xk |2 <∞.

`2 is a real or complex vector space. The inner product is

defined by 〈(xk), (yk)〉 =∞∑

k=0

xkyk


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Little `2


k=0

|xk |2 <∞.



k=0

xkyk

Since `2 is complete with respect to the induced norm, it is areal or complex Hilbert space.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Little `2


k=0

|xk |2 <∞.



k=0

xkyk

Since `2 is complete with respect to the induced norm, it is areal or complex Hilbert space.

Note that `2 is an infinite dimensional Hilbert space and anorthonormal basis is {(xi )n}n∈N∪{0} : xi = 0 for all i 6= n andxi = 1 if i = n.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

The Hardy Space H2

Let T := {z : ‖z‖ = 1} be the complex unit circle and let µbe Lebesgue measure (think arclength) normalized so thatµ(T) = 1(instead of 2π).


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

The Hardy Space H2


If we define en(z) = zn for n ≥ 0, the space H2 is thesubspace of L2 spanned by the en’s.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

The Hardy Space H2



The functions of H2 are called the analytic elements of L2.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

The Hardy Space H2



The functions of H2 are called the analytic elements of L2.

Of course one must note that what we are really talkingabout are equivalence classes of functions and so thequalification ”almost everywhere” must be put on anystatements involving them.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

The Hardy Space cont. H2.

One can look at the Hardy Space as honest to goodnessfunctions in the following way.

I If f ∈ H2, with Fourier expansion f =∑∞

n=0 αnen, then∑∞n=0 ‖αn‖2 <∞, and thus the radius of convergence

of the power series∑∞

n=0 α2nzn is greater than or equal

to 1.

I It follows that the power series f =∑∞

n=0 αnzn definesan analytic function f in the open unit disc D.

I Thus the mapping f → f establishes a one-to-onecorrespondence between H2 and the set H2 of thosefunctions analytic in D whose series of Taylorcoefficients is square summable.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

The Hardy Space cont. H2 and H2

If we have two functions φ and ψ with φ(z) =∑∞

n=0 αnzn

and ψ(z) =∑∞

n=0 βnzn, then the inner product is given by:

〈φ, ψ〉 :=∞∑

n=0

αnβn.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

The Hardy Space cont. H2 and H2

If we have two functions φ and ψ with φ(z) =∑∞

n=0 αnzn

and ψ(z) =∑∞

n=0 βnzn, then the inner product is given by:

〈φ, ψ〉 :=∞∑

n=0

αnβn.

In view of the correspondence f → f it all comes to thesame thing. We can call f the extension of f to the interior.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

The Bergman Space A2

Let µ be planar Lebesgue measure(think area) inD := {z ∈ C : ‖z‖ < 1} and let A2(D) be the set of allcomplex-valued functions that are analytic throughout D andsquare integrable with respect to µ.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



The set A2(D) or simply A2 is a vector space with respect topointwise addition and scaler multiplication.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion




It is also an inner product space with respect to

〈f , g〉 =

∫D

f (z)g(z)dµ(z).


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion




It is also an inner product space with respect to

〈f , g〉 =

∫D

f (z)g(z)dµ(z).

Since A2 is complete it is a Hilbert space.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Basis for the Bergman Space A2.

Let en(z) =

√n + 1

πzn for ‖z‖ < 1 and n = 0, 1, 2, .... Then

E = {en(z)} is an orthonormal basis for A2


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel Functions

Theorem (Riesz Representation Theorem)

Let H be a Hilbert space, let φ : H→ C be a functional,then ∃! a ∈ H such that φ(x) = 〈x , a〉 ∀x ∈ H.

I If H is a functional Hilbert space, over E , then thelinear functional φy : H→ F given by φy (f ) = f (y) onH is bounded for each y ∈ E .

Definition (Kernel Function)

Let F be a class of functions defined on E , forming a HilbertSpace(complex or real). The function K (x , y) of x and y inE is called a reproducing kernel (r.k.) of F if:

I For every y , K (x , y) as a function of x belongs to F .

I The reproducing property : for every y ∈ E and everyf ∈ F ,

f (y) = 〈f (x),K (x , y)〉x


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

How to Find the Kernel

TheoremIf {ej} is an orthonormal basis for a functional Hilbert spaceH, then the kernel function K of H is given by

K (x , y) =∑

j

ej(x)ej(y).


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

How to Find the Kernel

TheoremIf {ej} is an orthonormal basis for a functional Hilbert spaceH, then the kernel function K of H is given by

K (x , y) =∑

j

ej(x)ej(y).

This result holds because of the reproducing property ofK (x , y) and a result from Fourier analysis which states that

in a Hilbert space, for any f ∈ H, f =∑

j

〈f , ej〉ej when the

ej ’s form an orthonormal basis.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Proof of Theorem

Proof


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Proof of Theorem

Proof For a fixed y , Ky (x) ∈ H so:


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Proof of Theorem

Proof For a fixed y , Ky (x) ∈ H so:

Ky (x) =∑

j

〈Ky (x), ej(x)〉ej(x)

=∑

j

〈ej(x),Ky (x)〉ej(x)

=∑

j

ej(y)ej(x)

=∑

j

ej(x)ej(y)


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of `2.

We want to look at the evaluation functional φi : `2 → F fori ∈ N given by φi (xn) = xi . So what is the kernel functionK (i , j)?


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of `2 cont.

Since we are looking at `2 we want to think of the kernel asK (i , j) an infinite matrix. If we fix a j we get a particularcolumn of the matrix and that should be an element of `2.

I K (1, 2) =∑

j

ej(1)ej(2) but this is of course 0 since

ej(1) = 0 unless j = 1 and ej(2) = 0 unless j = 2.

I Thus we can see that K (i , j) =

{1 i = j

0 i 6= j


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of `2 cont.


So to calculate this we find each K (i , j). For instance:

I K (1, 2) =∑

j




{1 i = j

0 i 6= j


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of `2 cont.


So to calculate this we find each K (i , j). For instance:

I K (1, 2) =∑

j




{1 i = j

0 i 6= j

Thus

K (i , j) =

1 0 0 . . .0 1 0 . . .0 0 1 . . ....

. . .


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of `2 cont.

So how does K (i , j) illustrate the reproducing property:f (y) = 〈f (x),K (x , y)〉?


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of `2 cont.


Let f = (xn) be a sequence in `2. If we let a j be fixed, thenf (i) is really just the i th coordinate of (xn) that is xi . Thusf (i) = xi = 〈(xn),K (i , j)〉 = 〈(xn), {0, 0, ..., 1, 0, 0, ...}〉where the 1 occurs in the i th position.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of `2 cont.



But this is just∞∑

k=1

(xk)ei = xi .


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of `2 cont.



But this is just∞∑

k=1

(xk)ei = xi .

So we have the reproducing property, that is we can evaluatea sequence at the j th term by taking the inner product of thesequence with the kernel function with j fixed.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of H2

We want to find the kernel of the Hardy space, but we willconsider H2 since it is nicer to think of it as a space offunctions so we can use the theorem to find the kernel,which is known as the Szego Kernel. We have anorthonormal basis so we need to compute the following.

I For H2, we have E = {zn : n ≥ 0}.

I So K (x , y) =∞∑

n=0

xnyn =∞∑

n=0

(xy)n =1

1− xy.

I We can be sure the series converges since x , y ∈ Dimplies ‖xy‖ < 1.


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of A2

To find the kernel of A2, known as the Bergman kernel:


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of A2

To find the kernel of A2, known as the Bergman kernel:Recall the orthonormal basis is

en(z) =

√n + 1

πzn for ‖z‖ < 1 and n = 0, 1, 2, ....


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Kernel of A2

To find the kernel of A2, known as the Bergman kernel:To find the kernel we evaluate:

K (x , y) =∞∑

n=0

(√n + 1

πxn

)(√n + 1

πyn

)

=∞∑

n=0

n + 1

πxnyn

=∞∑

n=0

n + 1

π(xy)n

=1

π

∞∑n=0

(n + 1)(xy)n

=1

π

(1

1− xy

)2


Spaces

Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Uniqueness of Kernels.

Let F denote a class of functions f (x) defined in E , forminga Hilbert Space with the norm ‖f ‖ and inner product 〈f1, f2〉.Let K (x , y) denote the corresponding reproducing kernel.

I Uniqueness: If a reproducing kernel exists, it is unique.

I In fact, if another K ′(x , y) existed we would have forsome y :


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Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Uniqueness of Kernels.

Let F denote a class of functions f (x) defined in E , forminga Hilbert Space with the norm ‖f ‖ and inner product 〈f1, f2〉.Let K (x , y) denote the corresponding reproducing kernel.

I Uniqueness: If a reproducing kernel exists, it is unique.

I In fact, if another K ′(x , y) existed we would have forsome y :

0 <∣∣K (x , y)− K ′(x , y)

∣∣2 = 〈K − K ′,K − K ′〉= 〈K − K ′,K 〉 − 〈K − K ′,K ′〉= 0


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Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Existence of Kernels.

For the existence of a reproducing kernel, K (x , y) it isnecessary and sufficient that for every y ∈ E , φy (f ) be acontinuous functional of f ∈ F .

I In fact if K exists, then‖φy (f )‖ ≤ ‖f ‖ 〈K (x , y),K (x , y)〉1/2 = K (y , y)1/2 ‖f ‖.

I In the other direction, if φy (f ) is a continuousfunctional, then by the Riesz Representation theorem,there is a function gy (x) ∈ F such thatf (y) = 〈f (x), gy (x)〉 and letting K (x , y) = gy (x) weget a reproducing kernel.


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Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Completion of an Incomplete Hilbert Space

There are times when we want to complete an incompletefunctional space in order to make a Hilbert space.

TheoremConsider a class of functions F forming an incompleteHilbert space. In order that there exist a functionalcompletion of the class it is necessary and sufficient that:

I 1*. For every fixed y ∈ E the linear functional φy (f )defined on F is bounded.

I 2*. For a Cauchy sequence {fm} ⊂ F , the conditionfm(y)→ 0 for every y implies ‖fm‖ → 0.


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Thomas Clark

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Introduction

Definitions

Examples

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H2 and H2

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion


There are times when we want to complete an incompletefunctional space in order to make a Hilbert space.We would like to do this in such a way so that the elementsadded to the space are functions as well, instead of idealelements like in the case of L2. The following theorem is themain result in this direction.





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Thomas Clark

Outline

Introduction

Definitions

Examples

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H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion


There are times when we want to complete an incompletefunctional space in order to make a Hilbert space.We would like to do this in such a way so that the elementsadded to the space are functions as well, instead of idealelements like in the case of L2. The following theorem is themain result in this direction.




If the functional completion is possible, it is unique.


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Thomas Clark

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Introduction

Definitions

Examples

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Proof of Theorem: Necessity

Proof


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Introduction

Definitions

Examples

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Kernel of `2

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Conclusion


ProofWe can see that statement 1* is necessary by noting that inthe existence theorem of reproducing kernels, it is implied,and the completed class would necessarily have such a kernel.


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Thomas Clark

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Introduction

Definitions

Examples

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H2 and H2

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



The necessity of statement 2* follows from the fact that aCauchy sequence in F is strongly convergent in the completespace to a function f , and the function f is the limit of fm atevery point y ∈ E .


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Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



The necessity of statement 2* follows from the fact that aCauchy sequence in F is strongly convergent in the completespace to a function f , and the function f is the limit of fm atevery point y ∈ E . Hence f = 0 so ‖f ‖ = 0 and the norms‖fm‖ → ‖f ‖ = 0.


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Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



The necessity of statement 2* follows from the fact that aCauchy sequence in F is strongly convergent in the completespace to a function f , and the function f is the limit of fm atevery point y ∈ E . Hence f = 0 so ‖f ‖ = 0 and the norms‖fm‖ → ‖f ‖ = 0.

Now to prove sufficiency.


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Thomas Clark

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Introduction

Definitions

Examples

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H2 and H2

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Proof of Theorem: Sufficiency

Let {fn} ⊂ F be a Cauchy sequence. For every fixed y , letMy be the bound of the functional φy (f ) so that

‖f (y)‖ ≤ My ‖f ‖


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Thomas Clark

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Introduction

Definitions

Examples

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Kernels

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Conclusion



‖f (y)‖ ≤ My ‖f ‖

Consequently, ‖fm(y)− fn(y)‖ ≤ My ‖fm − fn‖ .


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Thomas Clark

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Introduction

Definitions

Examples

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Kernel of `2

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Conclusion



‖f (y)‖ ≤ My ‖f ‖


It follows that {fn(y)} is a Cauchy sequence of complexnumbers and thus converges, so denote it by f (y). In thismanner every Cauchy sequence {fn} defines a function f towhich it converges at every x ∈ E .


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Thomas Clark

Outline

Introduction

Definitions

Examples

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H2 and H2

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Kernels

Kernel of `2

Kernel of H2

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Conclusion



‖f (y)‖ ≤ My ‖f ‖


It follows that {fn(y)} is a Cauchy sequence of complexnumbers and thus converges, so denote it by f (y). In thismanner every Cauchy sequence {fn} defines a function f towhich it converges at every x ∈ E .

Let F denote the new class of functions defined in this way,which certainly contains F , and give it norm:

‖f ‖1 = lim ‖fn‖

and note that the norm does not depend on choice of{fn} → f .


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Thomas Clark

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Introduction

Definitions

Examples

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Proof of Theorem: Sufficiency cont.

Now we need to see that F is complete and contains F as adense subset.


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Thomas Clark

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Introduction

Definitions

Examples

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion


Now we need to see that F is complete and contains F as adense subset.We certainly have the second half since F ⊂ F , the normscoincide for f ∈ F and every f ∈ F is by definition the limitof a cauchy sequence {fn} ⊂ F everywhere in E .


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Thomas Clark

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Introduction

Definitions

Examples

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



To prove completeness we let {fn} ⊂ F be a Cauchysequence. Since F is dense in F we can find a Cauchysequence {f ′n} ⊂ F such that

lim∥∥f ′n − fn

∥∥1

= 0.


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Thomas Clark

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Introduction

Definitions

Examples

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H2 and H2

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion



To prove completeness we let {fn} ⊂ F be a Cauchysequence. Since F is dense in F we can find a Cauchysequence {f ′n} ⊂ F such that

lim∥∥f ′n − fn

∥∥1

= 0.

Since {f ′n} → f ∈ F we can see that {fn} will be draggedalong to f as well.


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Thomas Clark

Outline

Introduction

Definitions

Examples

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H2 and H2

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Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Proof of Theorem: Uniqueness

We can see that F is unique since any f ∈ F ′ anothercompletion would have to be a limit of a Cauchy sequence inF and thus be in F . As the norm of f has to be lim ‖fn‖ itmust coincide with ‖f ‖1. Similarly anything in F would bein F ′.


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Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Proof of Theorem: Uniqueness

We can see that F is unique since any f ∈ F ′ anothercompletion would have to be a limit of a Cauchy sequence inF and thus be in F . As the norm of f has to be lim ‖fn‖ itmust coincide with ‖f ‖1. Similarly anything in F would bein F ′.Thus any functional completion of F must coincide with Fand have the same norm and scaler product as F .


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Thomas Clark

Outline

Introduction

Definitions

Examples

R2

`2

H2 and H2

A2

Kernels

Kernel of `2

Kernel of H2

Kernel of A2


Conclusion

Concluding Remarks.

I Thanks for coming!

I If you are interested in this topic feel free to see meafter class or sign up for Functional Analysis this fall.

Reproducing Kernels in Hilbert Spaces - math.unl.edujorr1/classes/2009summer/math896/... ·...

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Transcript of Reproducing Kernels in Hilbert Spaces - math.unl.edujorr1/classes/2009summer/math896/... ·...