View source for Reproducing kernel

Consider an abstract set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300801.png" /> and a linear set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300802.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300803.png" />.

Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300804.png" /> is equipped with an [[Inner product|inner product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300806.png" /> is complete with respect to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300807.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300808.png" /> is a [[Hilbert space|Hilbert space]].

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008010.png" />, is called a reproducing kernel of such a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008011.png" /> if and only if the following two conditions are satisfied:

i) for every fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008012.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008013.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008015.png" />.

This definition is given in [[#References|[a1]]]; see also [[#References|[a6]]].

Some properties of reproducing kernels are:

1) If a reproducing kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008016.png" /> exists, then it is unique.

2) A reproducing kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008017.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008020.png" />.

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008021.png" /> is a non-negative-definite kernel, that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008022.png" /></td> </tr></table>

where the overbar stands for complex conjugation.

In particular, 3) implies:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008023.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008024.png" /></td> </tr></table>

Every non-negative-definite kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008025.png" /> generates a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008026.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008027.png" /> is a reproducing kernel (see also [[Reproducing-kernel Hilbert space|Reproducing-kernel Hilbert space]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008028.png" /> is a reproducing kernel, then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008029.png" /> is injective: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008030.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008031.png" />, by reproducing property ii), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008032.png" /> is surjective (cf. also [[Injection|Injection]]; [[Surjection|Surjection]]). Therefore the inverse operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008033.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008034.png" />, and since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008035.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008036.png" /> is the identity operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008037.png" />, and so is its inverse.

==Examples of reproducing kernels.==


Consider the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008038.png" /> of analytic functions (cf. [[Analytic function|Analytic function]]) in a bounded [[Simply-connected domain|simply-connected domain]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008039.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008040.png" />-plane. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008041.png" /> is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008043.png" />, and the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008044.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008045.png" /></td> </tr></table>

Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008046.png" /> is a reproducing-kernel Hilbert space. Its reproducing kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008047.png" /> is called the Bergman kernel (cf. also [[Bergman kernel function|Bergman kernel function]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008048.png" /> is an orthonormal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008049.png" /> (cf. also [[Orthogonal system|Orthogonal system]]; [[Basis|Basis]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008051.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008052.png" /> is the [[Conformal mapping|conformal mapping]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008053.png" /> onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008054.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008056.png" />, then [[#References|[a2]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008057.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008058.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008060.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008061.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008062.png" /> is a finite [[Measure|measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008063.png" />.

Define a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008064.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008065.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

Define the kernel

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

This kernel is non-negative-definite:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008067.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008068.png" /></td> </tr></table>

provided that for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008069.png" /> the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008070.png" /> is linearly independent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008071.png" /> (cf. [[Linear independence|Linear independence]]).

In this case the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008072.png" /> generates a uniquely determined reproducing-kernel Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008073.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008074.png" /> is the reproducing kernel.

In [[#References|[a6]]] it is claimed that a convenient characterization of the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008075.png" /> of the linear transformation (a1) is given by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008076.png" />. In [[#References|[a4]]] it is shown by examples that such a characterization is often useless in practice: in general the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008077.png" /> can not be described in terms of the standard Sobolev or Hölder norms, and the assumption in [[#References|[a6]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008078.png" /> can be realized as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008079.png" /> is not justified and is not correct, in general.

However, in [[#References|[a6]]] there are some examples of characterizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008080.png" /> for some special operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008081.png" /> and in [[#References|[a5]]] a characterization of the range of a wide class of multi-dimensional linear transforms, whose kernels are kernels of positive rational functions of self-adjoint elliptic operators, is given.

Reproducing kernels are discussed in [[#References|[a5]]] for rigged triples of Hilbert spaces (cf. also [[Rigged Hilbert space|Rigged Hilbert space]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008082.png" /> is a Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008083.png" /> is a linear [[Compact operator|compact operator]] defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008084.png" />, then the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008085.png" /> in the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008086.png" /> is a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008087.png" />. The space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008088.png" />, with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008089.png" />, is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008091.png" />. The inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008092.png" /> is given by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008093.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008094.png" />, equipped with this inner product, is a Hilbert space.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008095.png" />, where the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008096.png" /> are counted according to their multiplicities and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008097.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008098.png" /> is the Kronecker delta.

Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008099.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080100.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080101.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080102.png" />.

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080103.png" /> is a reproducing-kernel Hilbert space and its reproducing kernel is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080104.png" />.

To check that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080105.png" /> is indeed the reproducing kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080106.png" />, one calculates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080107.png" />. Indeed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080108.png" /> is the identity operator because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080109.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080110.png" /> is the kernel of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080111.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080112.png" />.

The value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080113.png" /> is a linear functional in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080114.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080115.png" /> is a reproducing-kernel Hilbert space. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080116.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080117.png" />. Therefore, denoting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080118.png" /> and using the [[Cauchy inequality|Cauchy inequality]] and [[Parseval equality|Parseval equality]] one gets:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080119.png" /></td> </tr></table>

as claimed.

From the representation of the inner product in the reproducing-kernel Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080120.png" /> by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080121.png" /> it is clear that, in general, the inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080122.png" /> is not an inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080123.png" />.

The inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080124.png" /> is of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080125.png" /></td> </tr></table>

where the distributional kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080126.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080127.png" /> by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080128.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080129.png" /> is the Fourier coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080130.png" /> (cf. also [[Fourier coefficients|Fourier coefficients]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080131.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080132.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080133.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080134.png" />. Thus, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080135.png" /> converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080136.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Aronszajn,   "Theory of reproducing kernels"  ''Trans. Amer. Math. Soc.'' , '''68'''  (1950)  pp. 337–404</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Bergman,   "The kernel function and conformal mapping" , Amer. Math. Soc.  (1950)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.G. Ramm,   "On the theory of reproducing kernel Hilbert spaces"  ''J. Inverse Ill-Posed Probl.'' , '''6''' :  5  (1998)  pp. 515–520</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Ramm,   "On Saitoh's characterization of the range of linear transforms"  A.G. Ramm (ed.) , ''Inverse Problems, Tomography and Image Processing'' , Plenum  (1998)  pp. 125–128</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.G. Ramm,   "Random fields estimation theory" , Longman/Wiley  (1990)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Saitoh,   "Integral transforms, reproducing kernels and their applications" , ''Pitman Res. Notes'' , Longman  (1997)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Schwartz,   "Sous-espaces hilbertiens d'espaces vectoriels topologique et noyaux associes"  ''Anal. Math.'' , '''13'''  (1964)  pp. 115–256</TD></TR></table>