Reproducing kernel Hilbert space: Difference between revisions

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In [[functional analysis]] (a branch of [[mathematics]]), a '''reproducing kernel Hilbert space''' ('''RKHS''') is a [[Hilbert space]] of functions in which point evaluation is a continuous linear [[Functional (mathematics)|functional]]. Roughly speaking, this means 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>|f(x)-g(x)|</math> is small for all <math>x</math>. The converse does not need to be true. Informally, this can be shown by looking at the [[Uniform norm|supremum norm]]: the sequence of functions <math>\sin^n (x)</math> converges pointwise, but does not converge [[Uniform Convergence|uniformly]] i.e. does not converge with respect to the supremum norm. (This is not a counterexample because the supremum norm does not arise from any [[inner product]] due to not satisfying the [[Polarization identity|parallelogram law]].)

ByIt theis abovenot definition,entirely anstraightforward exampleto ofconstruct a Hilbert space of functions thatwhich is not aan RKHS are the functions in .<mathref>C^\infty[aAlpay,b]</math> with a uniform bound <math>|f| \leq M</math>D., and withT. theM. usual inner productMills. This"A sincefamily aof sufficientlyHilbert narrowspaces bumpwhich functionare <math>f</math>not peakingreproducing atkernel <math>x</math>Hilbert canspaces." haveJ. arbitrarilyAnal. smallAppl. <math>\int |f|^1.2</math> while <math>f(x2003): 107–111.</mathref> remainsSome boundedexamples, below.however, Seehave 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 Szeg¨o type", ''Journal of Contemporary Mathematical Analysis'' (Armenian Academy of Sciences), 55, 2020. </ref><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>{{Failed verification|date=January 2024}}

[[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).