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Tito Omburo (talk | contribs) →Example: Paley-Wiener theorem |
Tito Omburo (talk | contribs) →Example: a simpler example |
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for every <math> n \in \mathbb{N}, x_1, \dots, x_n \in X, \text{ and } c_1, \dots, c_n \in \mathbb{R}. </math><ref>Durrett</ref> The Moore–Aronszajn theorem (see below) is a sort of converse to this: if a function <math>K</math> satisfies these conditions then there is a Hilbert space of functions on <math>X</math> for which it is a reproducing kernel.
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The simplest example of a reproducing kernel Hilbert space is the space <math>L^2(X,\mu)</math> where <math>X</math> is a set and <math>X</math> is the [[counting measure]] on <math>X</math>. For <math>x\in X</math>, the reproducing kernel <math>K_x</math> is the [[indicator function]] of the one point set <math>x\in X</math>.
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▼
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:<math> H = \{ f \in L^2(\mathbb{R}) \mid \operatorname{supp}(F) \subset [-a,a] \} </math>
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: <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(\mathbb R)</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
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