Plancherel–Polya-type inequalities for entire functions of exponential type in Lp(Rd)

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 Rd . As an application it is shown that a function f ∈ Lp(Rd ), 1 p ∞, which is an entire function of exponential type is uniquely determined by a set of numbers {Ψj (f )}, j ∈ N, where {Ψj }, 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 {Ψj (f )}, j ∈ N, is given. This method is a generalization of the classical result of Duffin–Schaeffer about exponential frames on intervals. © 2006 Elsevier Inc. All rights reserved.