Functions with disconnected spectrum : sampling, interpolation, translates
Author(s)
Bibliographic Information
Functions with disconnected spectrum : sampling, interpolation, translates
（University lecture series, v. 65）
American Mathematical Society, c2016
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
Includes bibliographical references (p. 133-138)
Description and Table of Contents
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.
Table of Contents
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
by "Nielsen BookData"