Bumps and polynomials in RKHSs of translation-invariant kernels

Abstract

We use the uncertainty principle of harmonic analysis and the Beurling-Malliavin theorem to describe when a reproducing kernel Hilbert space of a translation-invariant positive-definite kernel contains bump functions and polynomials. How fast the spectral density of the kernel decays controls whether or not these functions are included in the Hilbert space.