Softcover ISBN:  9781470428891 
Product Code:  ULECT/65 
List Price:  $44.00 
MAA Member Price:  $39.60 
AMS Member Price:  $35.20 
Electronic ISBN:  9781470432164 
Product Code:  ULECT/65.E 
List Price:  $44.00 
MAA Member Price:  $39.60 
AMS Member Price:  $35.20 

Book DetailsUniversity Lecture SeriesVolume: 65; 2016; 138 ppMSC: Primary 41; 42; 94;
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, highdimensional and nonanalytic 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.ReadershipGraduate students and research mathematicians interested in harmonic analysis and signal theory.

Table of Contents

Chapters

Lecture 1. Orthogonal bases and frames

Lecture 2. Paley–Wiener and Bernstein spaces

Lecture 3. Beurling’s sampling theorem

Lecture 4. Interpolation

Lecture 5. Disconnected spectrum

Lecture 6. Universal sampling

Lecture 7. Sampling bounds

Lecture 8. Approximation of discrete functions and size of spectrum

Lecture 9. Highdimensional phenomena

Lecture 10. Unbounded spectra

Lecture 11. Almost integer translates

Lecture 12. Discrete translates in $L^p(\mathbb {R})$


Additional Material

Reviews

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 musthave 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


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

Chapters

Lecture 1. Orthogonal bases and frames

Lecture 2. Paley–Wiener and Bernstein spaces

Lecture 3. Beurling’s sampling theorem

Lecture 4. Interpolation

Lecture 5. Disconnected spectrum

Lecture 6. Universal sampling

Lecture 7. Sampling bounds

Lecture 8. Approximation of discrete functions and size of spectrum

Lecture 9. Highdimensional phenomena

Lecture 10. Unbounded spectra

Lecture 11. Almost integer translates

Lecture 12. Discrete translates in $L^p(\mathbb {R})$

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 musthave 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