Sampling and interpolation in Hilbert spaces of entire functions

Abstract

Sampling theory is the study of spaces of functions which are reconstructible from their values at certain sets of points, which gives rise to a sampling formula for the underlying space. For this, it is necessary to consider spaces of functions whose values at a set of points are well-defined. In this work we consider the sampling and interpolation problems in reproducing kernel Hilbert space of entire functions, called de Branges space. Functions in such space are square integrable on the real line with respect to some weight function, and satisfying some growth conditions. Some sampling and interpolation results in the Paley-Wiener spaces, which are a primary example of de Branges spaces, are reviewed. We develop necessary conditions for sampling and interpolating sequences which generalize some well-known sampling and interpolation results in the Paley-Wiener space. The proofs of the necessary conditions rely very much on the Homogeneous Approximation Property and the Comparison Theorem that we prove in de Branges space. We also give necessary and sufficient conditions for Plancherel-Polya sequences, and sufficient conditions for interpolation in some de Branges spaces of exponential type.