Reproducing kernel Hilbert space: Difference between revisions

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==Examples==

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\mu</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\}\subset X</math>.

Nontrivial reproducing kernel Hilbert spaces often involve [[analytic function]]s, as we now illustrate by example. Consider the Hilbert space of [[Bandlimiting|bandlimited]] [[continuous function]]s <math>H</math>. Fix some [[cutoff frequency]] <math> 0<a < \infty </math> and define the Hilbert space