De Branges Spaces as Models for a General Theory of Function Spaces

Clemson University, Clemson SC

Investigators

Abstract

Modern processing of signals is almost always performed on the digital (discretized) version of the original signals, which is obtained by sampling the signals on a discrete set. One of the fundamental issues when converting an analog (continuous) signal to a digital (discrete) one is the following question: Can the original signals be recovered from the samples, and if so, how accurately? The answer, of course, depends heavily on the nature of the signals that are being processed. For example, signals that are more complex (oscillatory) in nature require more samples for accurate reconstruction. The precise mathematical relationship between the rate of oscillation and the required rate of sampling is surprisingly delicate, especially when the samples are non-uniform. This project is centered on the problem of understanding precisely this relationship, including several other mathematical questions that arise naturally from it. Analytic function spaces have always played an important role in the mathematical theory of signal processing. One natural class of such spaces that is particularly useful when studying non-uniform sampling is the class of de Branges spaces. Introduced in the sixties, the theory of de Branges spaces encompassed a great deal of mathematical analysis knowledge at that time, and it continues to play an important role in modern mathematics, providing a setting for the interplay of various areas of mathematics, including Fourier analysis, spectral theory, operator theory, random matrix theory, analytic number theory, and mathematical physics. The main research objective of this project is to conduct a deeper analysis of de Branges function spaces, and use the resulting findings to attack and resolve several long-standing open problems. Many of these problems are much more general in nature and go far beyond the setting of de Branges spaces. The reason that de Branges spaces serve as a model rests on the fact that this class of spaces already exhibits most of the key difficulties confronting signal processors. Another important goal of this project is to develop a theory that will unify the theory of classical function spaces, and use this unification as a guideline for developing new methods for resolving the remaining open questions about these spaces.