Sampling Theory in Fourier and Signal

Volume 1 in this series laid the mathematical foundations of samplingtheory; Volume 2 surveys the many applications of the theory both withinmathematics and in other areas of science. Topics range over a wide variety ofareas, and each application is given a modern treatment.

List of contributors

xiv

Applications of sampling theory to combinatorial analysis, stirling numbers, special functions and the Riemann zeta function

1

(37)

P. L. Butzer

M. Hauss

Introduction

1

(2)

Binomial coefficient function

3

(7)

Gauss summation formula

10

(2)

Stirling functions

12

(4)

Stirling functions and the Riemann zeta function ζ(s)

16

(4)

Euler formulae for ζ(s) and summation formulae involving the Hilbert transform

20

(12)

Sampling expansions of particular special functions

32

(6)

Sampling theory and the arithmetic Fourier transform

38

(18)

W. J. Walker

Historical development

38

(4)

The AFT algorithm

42

(3)

Aliasing properties of the DFT and AFT

45

(5)

The summability algorithm

50

(2)

Deletion properties of the AFT algorithm

52

(2)

Deletion properties of the summability algorithm

54

(2)

Derivative sampling---a paradigm example of multichannel methods

56

(22)

J. R. Higgins

Introduction

56

(1)

A brief survey of derivative sampling

57

(3)

The Riesz basis method in its multichannel setting

60

(6)

Applications to constructive function theory

66

(8)

Multidimensional derivative sampling

74

(4)

Computational methods in linear prediction for band-limited signals based on past samples

78

(18)

D. H. Mugler

Introduction

78

(1)

Asymptotic expansions

79

(2)

An optimal set of prediction coefficients

81

(2)

An integral equation

83

(2)

Size of the optimal prediction coefficients

85

(1)

Alternative computational methods

86

(2)

Suppressing the effects of noise on the prediction

88

(4)

Non-uniformly spaced samples

92

(2)

Conclusion

94

(2)

Interpolation and sampling theories and linear ordinary boundary value problems

96

(34)

W.N. Everitt

G. Nasri-Roudsari

Introduction

96

(10)

Second-order problems

106

(14)

Shannon-type interpolation formulae

120

(7)

Higher-order problems

127

(3)

Sampling by generalized kernels

130

(28)

R.L. Stens

Introduction

130

(2)

Notation and preliminary results

132

(4)

Approximation of non-band-limited functions

136

(13)

Approximation of band-limited functions

149

(9)

Sampling theory and wavelets

158

(29)

A. Fischer

Introduction

158

(4)

The continous wavelet transform

162

(6)

Interpolation on shift-invariant spaces

168

(5)

Multiresolution analysis

173

(4)

Sampling approximation

177

(10)

Approximation by translates of a radial function

187

(22)

N. Dyn

Introduction

187

(4)

Quasi-interpolation on regular grids

191

(7)

Optimal approximation schemes on regular grids

198

(3)

Approximation on quasi-uniformly scattered centres

201

(8)

Almost sure sampling restoration of band-limited stochastic signals

209

(24)

T. Pogany

Historical overview and oversampling

209

(5)

The Piranashvili-Lee theory

214

(5)

The Gaposhkin-Klesov theory

219

(3)

Irregular-derivative Kotel' nikov series

222

(3)

Derivative sampling and the Gaposhkin theory

225

(6)

Gaposhkin theory and irregular sampling

231

(2)

Abstract harmonic analysis and the sampling theorem

233

(33)

M. M. Dodson

M. G. Beaty

Introduction

233

(1)

Locally compact abelian groups

234

(8)

Fourier theory

242

(7)

An abstract sampling theorem

249

(9)

Some applications

258

(8)

References

266

(23)

Author index

289

(4)

Subject index

293

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Sampling Theory in Fourier and Signal Analysis Volume 2: Advanced Topics

Summary

Table of Contents

Supplemental Materials