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Functions with Disconnected Spectrum: Sampling, Interpolation, Translates

Alexander M. Olevskii Tel Aviv University, Tel Aviv, Israel
Alexander Ulanovskii Stavanger University, Stavanger, Norway
Available Formats:
Softcover ISBN: 978-1-4704-2889-1
Product Code: ULECT/65
List Price: $44.00 MAA Member Price:$39.60
AMS Member Price: $35.20 Electronic ISBN: 978-1-4704-3216-4 Product Code: ULECT/65.E List Price:$44.00
MAA Member Price: $39.60 AMS Member Price:$35.20
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List Price: $66.00 MAA Member Price:$59.40
AMS Member Price: $52.80 Click above image for expanded view Functions with Disconnected Spectrum: Sampling, Interpolation, Translates Alexander M. Olevskii Tel Aviv University, Tel Aviv, Israel Alexander Ulanovskii Stavanger University, Stavanger, Norway Available Formats:  Softcover ISBN: 978-1-4704-2889-1 Product Code: ULECT/65  List Price:$44.00 MAA Member Price: $39.60 AMS Member Price:$35.20
 Electronic ISBN: 978-1-4704-3216-4 Product Code: ULECT/65.E
 List Price: $44.00 MAA Member Price:$39.60 AMS Member Price: $35.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$66.00 MAA Member Price: $59.40 AMS Member Price:$52.80
• Book Details

University Lecture Series
Volume: 652016; 138 pp
MSC: 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, 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.

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. High-dimensional phenomena
• Lecture 10. Unbounded spectra
• Lecture 11. Almost integer translates
• Lecture 12. Discrete translates in $L^p(\mathbb {R})$

• 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 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 652016; 138 pp
MSC: 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, 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.

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. High-dimensional 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 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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