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Toeplitz approach to problems of the uncertainty principle / Alexei Poltoratski.

By: Contributor(s): Material type: TextTextSeries: CBMS regional conference series in mathematics ; no 121.Publication details: Providence : AMS, 2015.Description: vii, 216 p. ; 26 cmISBN:
  • 9781470420178 (pbk. : acidfree paper)
Subject(s): DDC classification:
  • 510 23 C748
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 -- Bibliography.
Summary: 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.
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Includes bibliographical references.

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 --
Bibliography.

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.

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