**University Lecture Series**

Volume: 65;
2016;
138 pp;
Softcover

MSC: Primary 41; 42; 94;

Print ISBN: 978-1-4704-2889-1

Product Code: ULECT/65

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

**Electronic ISBN: 978-1-4704-3216-4
Product Code: ULECT/65.E**

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

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#### Supplemental Materials

# Functions with Disconnected Spectrum: Sampling, Interpolation, Translates

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*Alexander M. Olevskii; Alexander Ulanovskii*

The classical sampling problem is to
reconstruct entire functions with given spectrum \(S\) from
their values on a discrete set \(L\). From the geometric point
of view, the possibility of such reconstruction is equivalent to
determining for which sets \(L\) the exponential system with
frequencies in \(L\) forms a frame in the space
\(L^2(S)\). The book also treats the problem of interpolation
of discrete functions by analytic ones with spectrum in \(S\)
and the problem of completeness of discrete translates. The size and
arithmetic structure of both the spectrum \(S\) and the
discrete set \(L\) play a crucial role in these problems.

After an elementary introduction, the authors give a new
presentation of classical results due to Beurling, Kahane, and
Landau. The main part of the book focuses on recent progress in the
area, such as construction of universal sampling sets,
high-dimensional and non-analytic phenomena.

The reader will see how methods of harmonic and complex analysis
interplay with various important concepts in different areas, such as
Minkowski's lattice, Kolmogorov's width, and Meyer's
quasicrystals.

The book is addressed to graduate students and researchers
interested in analysis and its applications. Due to its many
exercises, mostly given with hints, the book could be useful for
undergraduates.

#### Readership

Graduate students and research mathematicians interested in harmonic analysis and signal theory.

#### Reviews & Endorsements

The style of exposition is clear and concise. Many proofs are given in the form of (challenging) exercises which explains the relatively small number of pages of the book in comparison to its extensive content. However, the book is an excellent guide to the literature, comprising not only recent but also old and obscure sources from a variety of related fields. It is a must-have for any researcher working in theoretical signal analysis and can be inspiring for every complex, harmonic or functional analyst.

-- Gunter Semmler, Mathematical Reviews

The book is written in a clear manner with systematic and careful citation of references. Each lecture contains exercises with hints proposed to the reader. It can be used by graduate and PhD students interested in acquiring not only classical results but also recent ones with possible applications in various areas.

-- Liviu Goras, Zentralblatt Math

#### Table of Contents

# Table of Contents

## Functions with Disconnected Spectrum: Sampling, Interpolation, Translates

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Lecture 1. Orthogonal bases and frames 112
- Lecture 2. Paley–Wiener and Bernstein spaces 1324
- Lecture 3. Beurling’s sampling theorem 2334
- Lecture 4. Interpolation 3546
- Lecture 5. Disconnected spectrum 4556
- Lecture 6. Universal sampling 5768
- Lecture 7. Sampling bounds 6980
- Lecture 8. Approximation of discrete functions and size of spectrum 7788
- Lecture 9. High-dimensional phenomena 8798
- Lecture 10. Unbounded spectra 97108
- Lecture 11. Almost integer translates 111122
- Lecture 12. Discrete translates in 𝐿^{𝑝}(ℝ) 123134
- Bibliography 133144
- Back Cover Back Cover1152