**CBMS Regional Conference Series in Mathematics**

Volume: 121;
2015;
216 pp;
Softcover

MSC: Primary 30; 33; 34; 42;

Print ISBN: 978-1-4704-2017-8

Product Code: CBMS/121

List Price: $49.00

Individual Member Price: $39.20

**Electronic ISBN: 978-1-4704-2262-2
Product Code: CBMS/121.E**

List Price: $49.00

Individual Member Price: $39.20

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#### Supplemental Materials

# Toeplitz Approach to Problems of the Uncertainty Principle

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*Alexei Poltoratski*

A co-publication of the AMS and CBMS

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.

A co-publication of the AMS and CBMS.

#### Table of Contents

# Table of Contents

## Toeplitz Approach to Problems of the Uncertainty Principle

- Cover Cover11 free
- Title page i2 free
- Contents v6 free
- Thanks vii8 free
- Mathematical shapes of uncertainty 110 free
- Gap theorems 1726
- A problem by Pólya and Levinson 6776
- Determinacy of measures and oscillations of high-pass signals 7786
- Beurling-Malliavin and Bernstein’s problems 95104
- The Type Problem 119128
- Toeplitz approach to UP 143152
- Toeplitz version of the Beurling-Malliavin theory 183192
- Bibliography 211220
- Back Cover Back Cover1226

#### Readership

Graduate students and research mathematicians interested in harmonic and complex analysis and special problems.