Reproducing kernel Hilbert space: Difference between revisions

Revision as of 02:07, 13 May 2004

Revision as of 02:39, 13 May 2004

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In [[mathematics]] and [[functional analysis]] a '''reproducing kernel Hilbert space''' is a [[function space]] in which pointwise evaluation is a continuous linear functional. Alternatively, we will show they are spaces that can be defined by [[reproducing kernels]]. The subject was originally and simultaneously developed by [[N. Aronszajn]] and [[S. Bergman]] in [[1950]].

In this article we assume that Hilbert spaces are complex. This is becuasebecause 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

:<math> f \mapsto f(x) </math>

is norm continuous for any element ''x'' of ''X''. By the [[Riesz representation theorem]], this implies that there exists an element ''K''_''x of ''H'' such that

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The function

:<math> K(x,y) = K_x(y) </math>

is called a reproducing kernel for the Hilbert space. In fact, ''K'' is uniquely determined. by the condition

:<math> f(x) = \langle K(x, \cdot), f(\cdot)) \rangle </math>