Plancherel–Polya-type inequalities for entire functions of exponential type in [formula omitted

Abstract

The goal of the paper is to prove generalizations of the classical Plancherel–Polya inequalities in which point-wise sampling of functions ( δ-distributions) is replaced by more general compactly supported distributions on R d . As an application it is shown that a function f ∈ L p ( R d ) , 1 ⩽ p ⩽ ∞ , which is an entire function of exponential type is uniquely determined by a set of numbers { Ψ j ( f ) } , j ∈ N , where { Ψ j } , j ∈ N , is a countable sequence of compactly supported distributions. In the case p = 2 a reconstruction method of a Paley–Wiener function f from a sequence of samples { Ψ j ( f ) } , j ∈ N , is given. This method is a generalization of the classical result of Duffin–Schaeffer about exponential frames on intervals.