Reproducing kernel Hilbert space: Difference between revisions

Revision as of 12:25, 7 July 2024

Revision as of 11:10, 27 August 2024

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The space of [[Bandlimiting|bandlimited]] [[continuous function]]s <math>H</math> is a RKHS, as we now show. Formally, fix some [[cutoff frequency]] <math> 0<a < \infty </math> and define the Hilbert space

:<math> H = \{ f \in CL^2(\mathbb{R}) \mid \operatorname{supp}(F) \subset [-a,a] \} </math>

where <math>CL^2(\mathbb{R})</math> is the set of continuous square integrable functions, and <math display="inline"> F(\omega) = \int_{-\infty}^\infty f(t) e^{-i\omega t} \, dt </math> is the [[Fourier transform]] of <math> f</math>. As the inner product of this Hilbert space, we use

: <math>\langle f, g\rangle_{L^2} = \int_{-\infty}^\infty f(x) \cdot \overline{g(x)} \, dx.</math>

Since this is a closed subspace of <math>L^2</math>, it is a HIlbert space. Moreover, the elements of <math>H</math> are smooth functions on <math>\mathbb R</math> that tend to zero at infinity, essentially by the [[Riemann-Lebesgue lemma]]. In fact, the elements of <math>H</math> are the restrictions to <math>\mathbb R</math> of entire [[holomorphic function]]s, by the [[Payley-Wiener theorem]].

From the [[Fourier inversion theorem]], we have