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‘’‘reproducing kernel Hilbert space'''space’’’ ('''RKHS'''’’‘RKHS’’’) is a [[Hilbert space]] of functions in which point evaluation is a continuous linear [[Functionalfunctional (mathematics)|functional]]. Roughly speakingSpecifically, thisa meansHilbert thatspace if two functions <math>f</math> and <math>g</math> in the RKHS are close in norm, i.e., <math>\|f-g\|</math>mathcal{H} is small,of thenfunctions <math>f</math>from anda <math>g</math>set are alsoX pointwise close, i.e., <math>|f(x)-g(x)|</math> is small for all <math>x</math>. The converse does not need to be true.\mathbb{R} Informally, thisor can be shown by looking at the [[Uniform norm|supremum norm]]: the sequence of functions <math>\sin^mathbb{2nC} (x)</math> convergesis pointwise,an butRKHS doesif, notfor convergeeach [[Uniform Convergence|uniformly]]x i.e.\in doesX not, convergethere with respect to the supremum norm. (This is notexists a counterexamplefunction because thek_x supremum\in norm\mathcal{H} does notsuch arisethat fromfor anyall [[inner product]]f due\in to\mathcal{H} not satisfying the [[Polarization identity|parallelogram law]].),

\langle f, k_x \rangle = f(x).

It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS.<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>

The function k_x is called the reproducing kernel, and it reproduces the value of f at x via the inner product.

[[Square-integrable function|''L''² spaces]] are not Hilbert spaces of functions (and hence not RKHSs), but rather Hilbert spaces of equivalence classes of functions (for example, the functions <math>f</math> and <math>g</math> defined by <math>f(x)=0</math> and <math>g(x)=1_{\mathbb{Q}}</math> are equivalent in ''L''²). 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 immediate consequence of this property is that if two functions f and g in the RKHS are close in norm (i.e., \|f - g\| is small), then f and g are also pointwise close (i.e., \sup |f(x) - g(x)| 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 \sin^{2n}(x) . These functions converge pointwise to 0 as n \to \infty , 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 of functions which isare 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]] areis notusually defined as a Hilbert spacesspace whose elements are equivalence classes of functions (andit hencecan notbe RKHSs),trivially butredefined ratheras a Hilbert spacesspace of equivalencefunctions classesby ofusing functionschoice to select a (total) function as a representative for exampleeach equivalence class. However, theno functionschoice <math>f</math>of andrepresentatives <math>g</math>can definedmake bythis <math>f(x)=0</math>space andan RKHS (<math>g(x)=1_{\mathbb{Q}}k_0</math> arewould equivalentneed into ''L''²be 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.