Sampling Theory for Certain Hilbert Spaces of Bandlimited Functions

Abstract

With a given distribution Q positive-definite on an interval (-R, R), where 0 < R < ∞, and satisfying a certain non-degeneracy condition, one can associate a Hilbert space whose elements have well-defined Fourier transforms which are entire analytic functions. The problem of constructing various types of orthonormal bases, Riesz bases, frames, and normalized tight frames for these spaces is shown to be essentially equivalent to constructing certain positive tempered measures whose (distributional) inverse Fourier transforms agree with Q on (-R, R). A parametrization for such measures is then obtained.