Reproducing kernel Hilbert space: Difference between revisions

Revision as of 21:50, 28 September 2024

Revision as of 11:57, 21 October 2024

Line 67:

: <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 HIlbertHilbert 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 [[Paley–Wiener theorem]].

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