Reproducing kernel Hilbert space

In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a function space in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels. The subject was originally and simultaneously developed by Nachman Aronszajn (1907-1980) and Stephan Bergman (1895-1987) in 1950.

In this article we assume that Hilbert spaces are complex. This is because many of the examples of reproducing kernel Hilbert spaces are spaces of analytic functions. Also recall the sesquilinearity convention: the inner product is linear in the second variable.

Let X be an arbitrary set and H a Hilbert space of complex-valued functions on X. H is a reproducing kernel Hilbert space iff the linear map

${\displaystyle f\mapsto f(x)}$

from H to the complex numbers is continuous for any x in X. By the Riesz representation theorem, this implies that there exists an element K_x of H such that for every function f in the space,

${\displaystyle f(x)=\langle K_{x},f\rangle .}$

The function

${\displaystyle K(x,y)=K_{x}(y)}$

is called a reproducing kernel for the Hilbert space. In fact, K is uniquely determined by the above condition. In some concrete contexts this amounts to saying

${\displaystyle f(x)=\int _{\Omega }K(x,y)f(y)\,dy}$

for every f, where Ω is the appropriate ___domain, often the real numbers or Rⁿ.

Bergman kernel

The Bergman kernel is defined for open sets D in Cⁿ. Take the Hilbert H space of square-integrable functions, for the Lebesgue measure on D, that are holomorphic functions. The theory is non-trivial in such cases as there are such functions, which are not identically zero. Then H is a reproducing kernel space, with kernel function the Bergman kernel; this example, with n = 1, was introduced by Bergman in 1922.