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,
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where the overbar stands for complex conjugation.
In particular, 3) implies:
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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
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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:
![]() |
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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:
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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 |
Reproducing kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reproducing_kernel&oldid=50502