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About this book

The chapters of this volume are based on talks given at the eleventh international Sampling Theory and Applications conference held in 2015 at American University in Washington, D.C. The papers highlight state-of-the-art advances and trends in sampling theory and related areas of application, such as signal and image processing. Chapters have been written by prominent mathematicians, applied scientists, and engineers with an expertise in sampling theory. Claude Shannon’s 100th birthday is also celebrated, including an introductory essay that highlights Shannon’s profound influence on the field. The topics covered include both theory and applications, such as:Compressed sensingNon-uniform and wave samplingA-to-D conversionFinite rate of innovationComputational neuroscienceTime-frequency analysisOperator theoryMobile sampling issuesSampling: Theory and Applications is ideal for mathematicians, engineers, and applied scientists working in sampling theory or related areas.

Table of Contents

Frontmatter

Claude Shannon: American Genius

Abstract
Claude Shannon was born one hundred years ago. This essay is a salute to his genius.
J. Rowland Higgins

Reconstruction of Signals: Uniqueness and Stable Sampling

Abstract
The classical sampling problem is to reconstruct a continuous signal with a given spectrum S from its samples on a discrete set Λ. Through the Fourier transform, the problems ask for which sets of frequencies Λ is the exponential system
$$\displaystyle \{e^{i\lambda t},\, \lambda \in \varLambda \} $$
complete, or constitutes a frame in the space L 2 on a given set S of finite measure? When S is a single interval, these problems were essentially solved by A. Beurling, A. Beurling and P. Malliavin in terms of appropriate densities of the discrete set Λ. H. Landau extended the necessity of the density conditions in these results to the general bounded spectra. However, when S is a disconnected set, no sharp sufficient condition for sampling and completeness can be expressed in terms of the density of the set Λ. Not only the size, but also the arithmetic structure of Λ comes into the play. This paper gives a short introduction into the subject of sampling and related problems. We present both classical and recent result.
Alexander Olevskii, Alexander Ulanovskii

Sampling Theory in a Fourier Algebra Setting

Abstract
Sampling theory has been studied in a variety of function spaces and our purpose here is to develop the theory in a Fourier algebra setting. This aspect is not as well-known as it might be, possibly because the approximate sampling theorem, central to our discussion, had what could be called a mysterious birth and a confused adolescence. We also discuss functions of the familiar bandlimited and bandpass types, showing that they too have a place in this Fourier algebra setting. This paper combines an expository and historical treatment of the origins of exact and approximate sampling, including bandpass sampling, all in the Fourier algebra setting. It has two objectives. The first is to provide an accessible and rigorous account of this sampling theory. The various cases mentioned above are each discussed in order to show that the Fourier algebra, a Banach space, is a broad and natural setting for the theory. The second objective is to clarify the early development of approximate sampling in the Fourier algebra and to unravel its origins.
M. Maurice Dodson, J. Rowland Higgins

Sampling Series, Refinable Sampling Kernels, and Frequency Band Limited Functions

Abstract
We study sampling series and their relationship to frequency band limited functions. Motivated by the theories of multiresolution analyses and subdivision, particular attention is paid to sampling series whose kernels are refinable.
After developing the basic properties of the sampling kernels under study, we consider three families of such kernels: (1) damped cardinal sines, (2) the fundamental functions for cardinal spline interpolation, and (3) a family of compactly supported kernels defined in terms of their masks. The limiting kernel of each family is the cardinal sine. In the cases (1) and (2) we present results concerning the limiting behavior of the corresponding sampling series when the data samples {c n} have polynomial growth as n →±.
Wolodymyr R. Madych

Prolate Shift Frames and Sampling of Bandlimited Functions

Abstract
The Shannon sampling theorem can be viewed as a special case of (generalized) sampling reconstructions for bandlimited signals in which the signal is expressed as a superposition of shifts of finitely many bandlimited generators. The coefficients of these expansions can be regarded as generalized samples taken at a Nyquist rate determined by the number of generators and basic shift rate parameter. When the shifts of the generators form a frame for the Paley–Wiener space, the coefficients are inner products with dual frame elements. There is a tradeoff between time localization of the generators and localization of dual generators. The Shannon sampling theorem is an extreme manifestation in which the coefficients are point values but the generating sinc function is poorly localized in time. This work reviews and extends some recent related work of the authors regarding frames for the Paley–Wiener space generated by shifts of prolate spheroidal wave functions, and the question of tradeoff between localization of the generators and of the dual frames is considered.
Jeffrey A. Hogan, Joseph D. Lakey

A Survey on the Unconditional Convergence and the Invertibility of Frame Multipliers with Implementation

Abstract
The paper presents a survey over frame multipliers and related concepts. In particular, it includes a short motivation of why multipliers are of interest to consider, a review as well as extension of recent results, devoted to the unconditional convergence of multipliers, sufficient and/or necessary conditions for the invertibility of multipliers, and representation of the inverse via Neumann-like series and via multipliers with particular parameters. Multipliers for frames with specific structure, namely Gabor multipliers, are also considered. Some of the results for the representation of the inverse multiplier are implemented in Matlab-codes and the algorithms are described.
Diana T. Stoeva, Peter Balazs

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