Reproducing-kernel Hilbert space
Let be a Hilbert space of functions defined on an abstract set
.
Let denote the inner product and let
be the norm in
. The space
is called a reproducing-kernel Hilbert space if there exists a function
, the reproducing kernel, on
such that:
1) for any
;
2) for all
(the reproducing property). From this definition it follows that the value
at a point
is a continuous linear functional in
:
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The converse is also true. The following theorem holds: A Hilbert space of functions on a set is a reproducing-kernel Hilbert space if and only if
for all
.
By the Riesz theorem, the above assumption implies the existence of a linear functional such that
. By the construction, the kernel
is the reproducing kernel for
.
An example of a construction of a reproducing-kernel Hilbert space is the rigged triple of Hilbert spaces , which is defined as follows [a5] (cf. also Rigged Hilbert space). Let
be a Hilbert space of functions, let
be a linear densely defined self-adjoint operator on
,
(the eigenvalues
are counted according to their multiplicities) and assume that
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Define to be the Hilbert space with inner product
.
is the completion of
in the norm
. Let
be the dual space to
with respect to
. Then the inner product in
is defined by the formula
and
, equipped with the inner product
, is a Hilbert space.
Define , where the overline stands for complex conjugation. For any
, one has
. Indeed,
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Furthermore,
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so that is the reproducing kernel in
. Moreover
, where
is a constant independent of
. Indeed, if
and
, then
,
, and
.
Thus is a reproducing kernel Hilbert space with the reproducing kernel
defined above. If
is a function on
such that
![]() | (a1) |
then one can define a pre-Hilbert space of functions of the form
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The inner product of two functions from is defined by
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This definition makes sense because of (a1) and because of reproducing property 2). In particular, , as follows from (a1), and if
then
, as follows from property 2).
Indeed,
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Thus, if , then
and
, so
as claimed.
Denote by the completion of
in the norm
. Then
is a reproducing-kernel Hilbert space and
is its reproducing kernel.
A reproducing-kernel Hilbert space is uniquely defined by its reproducing kernel. Indeed, if is another reproducing-kernel Hilbert space with the same reproducing kernel
, then
and
is dense in
: If
,
for all
, then
. Using this and the equality
for all
, one can check that
and vice versa, so
, that is,
and
consist of the same set of elements. Moreover, the norms in
and
are equal. Indeed, take an arbitrary
and a sequence
,
. Then
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Thus, the norms in and
are equal, as claimed, and so are the inner products (by the polarization identity).
Define a linear operator ,
, where
and
is the range
of
, which will be equipped with the structure of a Hilbert space below:
![]() | (a2) |
Here, is a domain in
and
is a positive measure on
,
,
for all
, and it is assumed that
is injective, that is, the system
is total in
(cf. also Total set).
Define
![]() | (a3) |
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This kernel clearly satisfies condition (a1) and therefore is a reproducing kernel for the reproducing-kernel Hilbert space which it generates. Clearly
for all
. If
, that is,
,
, then
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if one equips with the inner product such that
. This requirement is formally equivalent to the following one:
, where
, so that the distributional kernel
is not the usual delta-function, but the one which acts by the rule
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and formally one has .
With the inner product , the linear set
becomes a Hilbert space:
![]() | (a4) |
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Thus, this inner product makes an isometric operator defined on all of
and makes
a (complete) Hilbert space, namely
, a reproducing-kernel Hilbert space. Since
is assumed injective, it follows that
is defined on all of
and, since
is complete in the norm
, one concludes that
is continuous (by the Banach theorem). Consequently,
is a co-isometry, that is,
, where
is the adjoint operator to
. If
, then one can write an inversion formula for the linear transform
similar to the well-known inversion formula for the Fourier transform. Formally one has:
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The space is the reproducing-kernel Hilbert space generated by kernel (a3) which is the reproducing kernel for
. The above formal inversion formulas may be of practical interest if the norm in
is a standard one. In this case the second formula should be suitably interpreted, since
is defined at
-almost all
.
In [a6] it is claimed that the characterization of the range of the linear operator , defined in (a3), can be given as follows:
, where
is the reproducing-kernel Hilbert space generated by kernel (a3).
However, in fact such a characterization does not give, in general, practically useful necessary and sufficient conditions for because the norm in
is not defined in terms of standard norms such as Sobolev or Hölder ones (see [a3], [a4], [a5]). However, when the norm in
is equivalent to a standard norm, the above characterization becomes efficient (see [a3], [a4], [a5], and also [a6]).
Many concrete examples of reproducing-kernel Hilbert spaces can be found in [a1], [a2] and [a6].
The papers [a1] and [a7] are important in this area, the book [a6] contains many references, while [a2] is an earlier book important for the development of the theory of reproducing-kernel Hilbert spaces.
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 topologiques et noyaux associès (noyaux reproduisants)" J. Anal. Math. , 13 (1964) pp. 115–256 |
Reproducing-kernel Hilbert space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reproducing-kernel_Hilbert_space&oldid=18349