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.