Toeplitz approach to problems of the uncertainty principle
By: Alexei Poltoratski
Contributors: National Science Foundation (U.S.)
Series: CBMS regional conference series in mathematics -- number 121
Publisher: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society; American Mathematical Society
Published: 2015
Place of publication: Providence, Rhode Island
Language: eng
Number of pages: 216
ISBN-13: 9781470420178
Copyright date: 2015
Subjects: Heisenberg uncertainty principle; Congresses; Functional analysis; Functions of a complex variable; Special functions (33-XX deals with the properties of functions as functions); Ordinary differential equations; Harmonic analysis on Euclidean spaces
Description: The Uncertainty Principle in Harmonic Analysis (UP) is a classical, yet rapidly developing, area of modern mathematics. Its first significant results and open problems date back to the work of Norbert Wiener, Andrei Kolmogorov, Mark Krein and Arne Beurling. At present, it encompasses a large part of mathematics, from Fourier analysis, frames and completeness problems for various systems of functions to spectral problems for differential operators and canonical systems. These notes are devoted to the so-called Toeplitz approach to UP which recently brought solutions to some of the long-standing problems posed by the classics. After a short overview of the general area of UP the discussion turns to the outline of the new approach and its results. Among those are solutions to Beurling's Gap Problem in Fourier analysis, the Type Problem on completeness of exponential systems, a problem by Pólya and Levinson on sampling sets for entire functions, Bernstein's problem on uniform polynomial approximation, problems on asymptotics of Fourier integrals and a Toeplitz version of the Beurling-Malliavin theory. One of the main goals of the book is to present new directions for future research opened by the new approach to the experts and young analysts.
Notes: "Partially supported by the National Science Foundation."

Based on the NSF-CBMS Regional Conference in the Mathematical Sciences on Uncertainty Principles in Harmonic Analysis : Gap and Type Problems, held at Clemson University, Clemson, South Carolina, August 12-16, 2013.

Includes bibliographical references (pages 211-216).