Reproducing kernel Hilbert space: Difference between revisions

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[[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 [[functionallinear (mathematics)|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, for each <math>x \in X</math> , there exists a function <math>K_x \in H</math> such that for all <math>f \in H</math> ,

<math>\langle f, K_x \rangle = f(x).</math>

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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>

While, formally, [[Square-integrable function|''L''² spaces]] is usuallyare defined as a Hilbert spacespaces whose elements areof equivalence classes of functions, itthis definition can trivially be triviallyextended redefined asto a Hilbert space of functions by using choice to selectchosing a (total) function as a representative for each equivalence class. However, no choice of representatives can make this space an RKHS (<math>K_0</math> would need to be the non-existent Dirac delta function). However, there are RKHSs in which the norm is an ''L''²-norm, such as the space of band-limited functions (see the example below).

An RKHS is associated with a kernel that reproduces every function in the space in the sense that for every <math>x</math> in the set on which the functions are defined, "evaluation at <math>x</math>" can be performed by taking an inner product with a function determined by the kernel. Such a ''reproducing kernel'' exists if and only if every evaluation functional is continuous.