Modern Sampling Theory

1. Front Matter

   Pages i-xvi

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2. Introduction

   1. Introduction

      • John J. Benedetto, Paulo J. S. G. Ferreira

      Pages 1-26

3. On the Transmission Capacity of the “Ether” and Wire in Electrocommunications

   1. On the Transmission Capacity of the “Ether” and Wire in Electrocommunications

      • V. A. Kotel’nikov

      Pages 27-45

4. Sampling, Wavelets, and the Uncertainty Principle

   1. Front Matter

      Pages 47-47

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   2. Wavelets and Sampling

      • Gilbert G. Walter

      Pages 49-71

   3. Embeddings and Uncertainty Principles for Generalized Modulation Spaces

      • J. A. Hogan, J. D. Lakey

      Pages 73-105

   4. Sampling Theory for Certain Hilbert Spaces of Bandlimited Functions

      • Jean-Pierre Gabardo

      Pages 107-134

   5. Shannon-Type Wavelets and the Convergence of Their Associated Wavelet Series

      • Ahmed I. Zayed

      Pages 135-152

5. Sampling Topics from Mathematical Analysis

   1. Front Matter

      Pages 153-153

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   2. Non-Uniform Sampling in Higher Dimensions: From Trigonometric Polynomials to Bandlimited Functions

      • Karlheinz Gröchenig

      Pages 155-171

   3. The Analysis of Oscillatory Behavior in Signals Through Their Samples

      • Rodolfo H. Torres

      Pages 173-192

   4. Residue and Sampling Techniques in Deconvolution

      • Stephen Casey, David Walnut

      Pages 193-218

   5. Sampling Theorems from the Iteration of Low Order Differential Operators

      • J. R. Higgins

      Pages 219-227

   6. Approximation of Continuous Functions by Rogosinski-Type Sampling Series

      • Andi Kivinukk

      Pages 229-244

6. Sampling Tools and Applications

   1. Front Matter

      Pages 245-245

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   2. Fast Fourier Transforms for Nonequispaced Data: A Tutorial

      • Daniel Potts, Gabriele Steidl, Manfred Tasche

      Pages 247-270

   3. Efficient Minimum Rate Sampling of Signals with Frequency Support over Non-Commensurable Sets

      • Cormac Herley, Ping Wah Wong

      Pages 271-291

   4. Finite-and Infinite-Dimensional Models for Oversampled Filter Banks

      • Thomas Strohmer

      Pages 293-315

   5. Statistical Aspects of Sampling for Noisy and Grouped Data

      • M. Pawlak, U. Stadtmüller

      Pages 317-342

   6. Reconstruction of MRI Images from Non-Uniform Sampling and Its Application to Intrascan Motion Correction in Functional MRI

      • Marc Bourgeois, Frank T. A. W. Wajer, Dirk van Ormondt, Danielle Graveron-Demilly

      Pages 343-363

Sampling is a fundamental topic in the engineering and physical sciences. This new edited book focuses on recent mathematical methods and theoretical developments, as well as some current central applications of the Classical Sampling Theorem. The Classical Sampling Theorem, which originated in the 19th century, is often associated with the names of Shannon, Kotelnikov, and Whittaker; and one of the features of this book is an English translation of the pioneering work in the 1930s by Kotelnikov, a Russian engineer. Following a technical overview and Kotelnikov's article, the book includes a wide and coherent range of mathematical ideas essential for modern sampling techniques. These ideas involve wavelets and frames, complex and abstract harmonic analysis, the Fast Fourier Transform (FFT),and special functions and eigenfunction expansions. Some of the applications addressed are tomography and medical imaging. Topics:. Relations between wavelet theory, the uncertainty principle, and sampling; . Multidimensional non-uniform sampling theory and algorithms;. The analysis of oscillatory behavior through sampling;. Sampling techniques in deconvolution;. The FFT for non-uniformly distributed data; . Filter design and sampling; . Sampling of noisy data for signal reconstruction;. Finite dimensional models for oversampled filter banks; . Sampling problems in MRI. Engineers and mathematicians working in wavelets, signal processing, and harmonic analysis, as well as scientists and engineers working on applications as varied as medical imaging and synthetic aperture radar, will find the book to be a modern and authoritative guide to sampling theory.

• Fourier transform
• Modulation
• Shannon
• communication
• fast Fourier transform
• fast Fourier transform (FFT)
• harmonic analysis
• signal analysis
• signal processing

"The introduction (Chapter 1) gives an excellent overview of the history and development of sampling theory. It shows that the WSK sampling theory has roots in many classical areas of mathematics, such as harmonic analysis, number theory, and interpolation theory. Many famous mathematicians, such as Cauchy, Borel, Hadamard, and de la Vallee-Poussin contributed directly or indirectly to its development. The introduction then proceeds to show how sampling theory is connected to more recent topics in mathematical analysis, such as wavelets, Gabor systems, density theorems, frames, and sampling in locally compact abelian groups."

—Mathematical Reviews

"Engineers and mathematicians working in wavelets, signal processing, and harmonic analysis, as well as scientists and engineers working on applications as varied as medical imaging and synthetic aperture radar, will find the book to be a modern and authoritative guide to sampling theory."

—Publicationes Mathematicae

• Department of Mathematics, University of Maryland, College Park, USA

  John J. Benedetto

• Departamento de Electronica e Telecommunicacoes, Universidade de Aveiro, Aveiro, Portugal

  Paulo J. S. G. Ferreira

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