Reproducing kernel Hilbert space: Difference between revisions

Revision as of 07:53, 27 September 2024

Revision as of 21:50, 28 September 2024

Line 2:

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

In [[functional analysis]], a ‘’‘reproducing'''reproducing kernel Hilbert space’’’space''' (’’‘RKHS’’’'''RKHS''') is a [[Hilbert space]] of functions in which point evaluation is a continuous linear [[functional (mathematics)|functional]]. Specifically, a Hilbert space \mathcal{<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 k_x<math>K_x \in \mathcal{H}</math> such that for all <math>f \in \mathcal{H}</math> ,

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

The function k_x<math>K_x</math> is 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 if two functions <math>f</math> and <math>g</math> in the RKHS are close in norm (i.e., <math>\|f - g\|</math> is small), then <math>f</math> and <math>g</math> are also pointwise close (i.e., <math>\sup |f(x) - g(x)|</math> is small). This follows from the fact that the inner product induces pointwise evaluation control. Roughly speaking, this means that closeness in the RKHS norm implies pointwise closeness, but the converse does not necessarily hold.

For example, consider the sequence of functions <math>\sin^{2n}(x) </math>. These functions converge pointwise to 0 as <math>n \to \infty</math> , but they do not converge uniformly (i.e., they do not converge with respect to the supremum norm). This illustrates that pointwise convergence does not imply convergence in norm. It is important to note that the supremum norm does not arise from any inner product, as it does not satisfy the [[Polarization identity|parallelogram law]].

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 [[Square-integrable function|''L''² spaces]] is usually defined as a Hilbert space whose elements are equivalence classes of functions it can be trivially redefined as a Hilbert space of functions by using choice to select a (total) function as a representative for each equivalence class. However, no choice of representatives can make this space an RKHS (<math>k_0K_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.