Completeness Problems in Harmonic Analysis and Spectral Theory

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

This project is in the area of Complex and Harmonic Analysis and applications to Spectral Theory and Mathematical Physics. It concerns open problems on completeness of complex exponentials in weighted spaces of square-integrable functions posted decades ago by such prominent mathematicians as Norman Levinson, Andrei Kolmogorov, Mark Krein, George Polya and Norbert Wiener. Three key problems in this area, the Beurling-Malliavin Problem, the Gap Problem and the Type Problem will be revisited in this project. The Beruling-Malliavin Problem was solved in the early sixties, although its numerous generalizations and applications remain open. A solution to the Gap Problem was recently suggested by the investigator. The solution brings up a number of new questions that will be considered in this project. The Type Problem, perhaps the most studied and important of the three, remains open. Recent partial results, produced via the approach used for the Gap Problem, suggest that a full solution may finally be within reach. The problem of approximating an arbitrary wave by combinations of simple waves, or "harmonics," belongs to the very foundations of Harmonic Analysis. Versions of this problem have been studied by many famous mathematicians. Despite considerable efforts, many of such questions remain open. Such problems have a number of important applications in Approximation Theory, Signal Processing, Prediction Theory, Spectral Theory of differential operators and Mathematical Physics. Further progress in this direction is the main goal of the project. Solutions to these problems could bring a satisfactory closure to the decades-long study of this area of Harmonic Analysis. Some of these problems remained open for more than half a century, and some of them were even considered "transcendental," i.e. not having a closed form solution. However, a recently developed approach, described in the main part of the proposal, has produced solutions to several of such problems and brought new optimism to the area. A research conference in the area of the project was recently held at Texas A&M University (January 2011). A survey of the main topics of the project will be included in the graduate Complex Analysis classes taught by the investigator at Texas A&M University. Two minicourses in the area of the project, directed at young researchers, will be given by the investigator at an international research program in 2011.