Generalized Fourier Frames in Terms of Balayage

Abstract

We develop a theory of non-uniform sampling in the context of the theory of frames for the settings of the short time fourier transform and pseudo-differential operators. Our theory is based on profound historical precedents including Beurling’s theory of balayage, emanating from the nineteenth century work of Christoffel and Poincare, the theory and results from spectral synthesis due to Wiener and Beurling and a host of the major harmonic analysts of the twentieth century, and the theory of sets of multiplicity, going back to Riemann and emerging fundamentally from the Russian school of harmonic analysis in the early twentieth century. Our results are meant to serve as the underpinnings for both theoretical and practical results in the realm of non-uniform sampling. They can also be compared with several other distinct forays into non-uniform sampling, including the settings of quasi-crystals and modulation spaces, where proofs for the latter setting require the analysis of convolution operators on the Heisenberg group. Our theory herein is the first step in which the ultimate goal is computational implementation for non-uniform sampling and its myriad applications, where balayage, spectral synthesis, and sets of multiplicity are computationally quantified. A critical component is to resurrect the formulation of balayage in terms of covering criteria.