Reproducing kernel Hilbert space: Difference between revisions

Revision as of 07:21, 29 April 2025

Revision as of 07:26, 29 April 2025

Line 2:

[[File:Different Views on RKHS.png|thumb|right|Figure illustrates related but varying approaches to viewing RKHS]]

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>\delta_xL_x:H\to\mathbb{C}</math>, <math>\delta_xL_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>.