Paley–Wiener subspace of vectors in a Hilbert space with applications to integral transforms

Abstract

The goal of this article is to introduce an analogue of the Paley–Wiener space of bandlimited functions, PW ω , in Hilbert spaces and then apply the general result to more specific examples. Guided by the role that the differentiation operator plays in some of the characterizations of the Paley–Wiener space, we construct a space of vectors using a self-adjoint operator D in a Hilbert space H, and denote this space by PW ω ( D ) . The article can be virtually divided into two parts. In the first part we show that the space PW ω ( D ) has similar properties to those of the space PW ω , including an analogue of the Bernstein inequality and the Riesz interpolation formula. We also develop a new characterization of the abstract Paley–Wiener space in terms of solutions of Cauchy problems associated with abstract Schrödinger equations. Finally, we prove two sampling theorems for vectors in PW ω ( D ) , one of which uses the notion of Hilbert frames and the other is based on the notion of variational splines in H. In the second part of the paper we apply our abstract results to integral transforms associated with singular Sturm–Liouville problems. In particular we obtain two new sampling formulas related to one-dimensional Schrödinger operators with bounded potential.