Reproducing kernel

Consider an abstract set and a linear set of functions .

Assume that is equipped with an inner product and is complete with respect to the norm . Then is a Hilbert space.

A function , , is called a reproducing kernel of such a Hilbert space if and only if the following two conditions are satisfied:

i) for every fixed , the function ;

ii) , .

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

Some properties of reproducing kernels are:

1) If a reproducing kernel exists, then it is unique.

2) A reproducing kernel exists if and only if , , where .

3) is a non-negative-definite kernel, that is,

where the overbar stands for complex conjugation.

In particular, 3) implies:

Every non-negative-definite kernel generates a Hilbert space for which is a reproducing kernel (see also Reproducing-kernel Hilbert space).

If is a reproducing kernel, then the operator is injective: implies , by reproducing property ii), and is surjective (cf. also Injection; Surjection). Therefore the inverse operator is defined on , and since , the operator is the identity operator on , and so is its inverse.

Examples of reproducing kernels.

Consider the Hilbert space of analytic functions (cf. Analytic function) in a bounded simply-connected domain of the complex -plane. If is analytic in , , and the disc , then

Therefore is a reproducing-kernel Hilbert space. Its reproducing kernel is called the Bergman kernel (cf. also Bergman kernel function).

If is an orthonormal basis of (cf. also Orthogonal system; Basis), , then .

If is the conformal mapping of onto the disc , such that , , then [a2]:

Let be a domain in and for every . Here is a finite measure on .

Define a linear mapping by

(a1)

Define the kernel

(a2)

This kernel is non-negative-definite:

provided that for any set the set of functions is linearly independent in (cf. Linear independence).

In this case the kernel generates a uniquely determined reproducing-kernel Hilbert space for which is the reproducing kernel.

In [a6] it is claimed that a convenient characterization of the range of the linear transformation (a1) is given by the formula . In [a4] it is shown by examples that such a characterization is often useless in practice: in general the norm in can not be described in terms of the standard Sobolev or Hölder norms, and the assumption in [a6] that can be realized as is not justified and is not correct, in general.

However, in [a6] there are some examples of characterizations of for some special operators and in [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 [a5] for rigged triples of Hilbert spaces (cf. also Rigged Hilbert space). If is a Hilbert space and is a linear compact operator defined on all of , then the closure of in the norm is a Hilbert space . The space dual to , with respect to , is denoted by , . The inner product in is given by the formula . The space , equipped with this inner product, is a Hilbert space.

Let , where the eigenvalues are counted according to their multiplicities and , where is the Kronecker delta.

Assume that for all and all , and .

Then is a reproducing-kernel Hilbert space and its reproducing kernel is .

To check that is indeed the reproducing kernel of , one calculates . Indeed, is the identity operator because , so that is the kernel of the operator in .

The value is a linear functional in , so that is a reproducing-kernel Hilbert space. Indeed, if , then . Therefore, denoting and using the Cauchy inequality and Parseval equality one gets:

as claimed.

From the representation of the inner product in the reproducing-kernel Hilbert space by the formula it is clear that, in general, the inner product in is not an inner product in .

The inner product in is of the form

where the distributional kernel acts on by the formula , where is the Fourier coefficient of (cf. also Fourier coefficients). If , then for some , and . Thus, the series converges in .

References

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