
Functions with Disconnected Spectrum
Sampling, Interpolation, Translates
American Mathematical Society (Publisher)
Will be published approx. on 30. June 2016
Book
Paperback/Softback
143 pages
978-1-4704-2889-1 (ISBN)
Description
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.
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.
Reviews / Votes
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"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
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Dimensions
Height: 254 mm
Width: 178 mm
Weight
278 gr
ISBN-13
978-1-4704-2889-1 (9781470428891)
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Schweitzer Classification
Persons
Alexander M. Olevskii, Tel Aviv University, Israel.
Alexander Ulanovskii, Stavanger University, Norway.
Alexander Ulanovskii, Stavanger University, Norway.
Content
Orthogonal bases and frames
Paley-Wiener and Bernstein spaces
Beurling's sampling theorem
Interpolation
Disconnected spectrum
Universal sampling
Sampling bounds
Approximation of discrete functions and size of spectrum
High-dimensional phenomena
Unbounded spectra
Almost integer translates
Discrete translates in $L^p(\mathbb{R})$
Bibliography
Paley-Wiener and Bernstein spaces
Beurling's sampling theorem
Interpolation
Disconnected spectrum
Universal sampling
Sampling bounds
Approximation of discrete functions and size of spectrum
High-dimensional phenomena
Unbounded spectra
Almost integer translates
Discrete translates in $L^p(\mathbb{R})$
Bibliography