Toeplitz Approach to Problems of the Uncertainty Principle
About this Title
Alexei Poltoratski, Texas A&M University, College Station, TX
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 2015: Volume 121
ISBNs: 978-1-4704-2017-8 (print); 978-1-4704-2262-2 (online)
MathSciNet review: MR3309830
MSC: Primary 30B60; Secondary 30D15, 30H15, 42-02
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.
Graduate students and research mathematicians interested in harmonic and complex analysis and special problems.
Table of Contents
- 1. Mathematical shapes of uncertainty
- 2. Gap theorems
- 3. A problem by Pólya and Levinson
- 4. Determinacy of measures and oscillations of high-pass signals
- 5. Beurling-Malliavin and Bernstein’s problems
- 6. The Type Problem
- 7. Toeplitz approach to UP
- 8. Toeplitz version of the Beurling-Malliavin theory