Reproducing kernel Hilbert space: Difference between revisions

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Latest revision as of 07:22, 17 August 2025

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In [[functional analysis]], a '''reproducing kernel Hilbert space''' ('''RKHS''') is a [[Hilbert space]] of functions in which point evaluation is a continuous [[linear functional]]. Specifically, a Hilbert space <math>H</math> of functions from a set <math>X</math> (to <math>\mathbb{R}</math> or <math>\mathbb{C}</math>) is an RKHS if the point-evaluation functional <math>L_x:H\to\mathbb{C}</math>, <math>L_x(f)=f(x)</math>, is continuous for every <math>x\in X</math>. Equivalently, <math>H</math> is an RKHS if there exists a function <math>K_x \in H</math> such that, for all <math>f \in H</math>,<math display="block">\langle f, K_x \rangle = f(x).</math>The function <math>K_x</math> is then called the ''reproducing kernel'', and it reproduces the value of <math>f</math> at <math>x</math> via the inner product.

An immediate consequence of this property is that convergence in norm implies [[uniform convergence]] on any subset of <math>X</math> on which <math>\|K_x\|</math> is bounded. However, the converse does not necessarily hold. Often the set <math>X</math> carries a topology, and <math>\|K_x\|</math> depends continuously on <math>x\in X</math>, in which case: convergence in norm implies uniform convergence on compact subsets of <math>X</math>.

It is not entirely straightforward to construct natural examples of a Hilbert space which are not an RKHS in a non-trivial fashion.<ref>Alpay, D., and T. M. Mills. "A family of Hilbert spaces which are not reproducing kernel Hilbert spaces." J. Anal. Appl. 1.2 (2003): 107–111.</ref> Some examples, however, have been found.<ref> Z. Pasternak-Winiarski, "On weights which admit reproducing kernel of Bergman type", ''International Journal of Mathematics and Mathematical Sciences'', vol. 15, Issue 1, 1992. </ref><ref> T. Ł. Żynda, "On weights which admit reproducing kernel of Szegő type", ''Journal of Contemporary Mathematical Analysis'' (Armenian Academy of Sciences), 55, 2020. </ref>

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:'''Theorem'''. Suppose ''K'' is a symmetric, [[positive definite kernel]] on a set ''X''. Then there is a unique Hilbert space of functions on ''X'' for which ''K'' is a reproducing kernel.

'''Proof'''. For all ''x'' in ''X'', define ''K_x'' = ''K''(''x'', ⋅ ). Let ''H''₀ be the [[linear span]] of {''K_x'' : ''x'' ∈ ''X''}. Define an inner product on ''H''₀ by

:<math> \left\langle \sum_{j=1}^n b_j K_{y_j}, \sum_{i=1}^m a_i K_{x_i} \right \rangle_{H_0} = \sum_{i=1}^m \sum_{j=1}^n {a_i} b_j K(y_j, x_i),</math>