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A Handbook of Fourier Theorems - D.C. Champeney

A Handbook of Fourier Theorems


Published: 1st May 1989
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This book is concerned with the well-established mathematical technique known as Fourier analysis (or alternatively as harmonic or spectral analysis). It is a handbook comprising a collection of the most important theorems in Fourier analysis, presented without proof in a form that is accurate but also accessible to a reader who is not a specialist mathematician. The technique of Fourier analysis has long been of fundamental importance in the physical sciences, engineering and applied mathematics, and is today of particular importance in communications theory and signal analysis. Existing books on the subject are either rigorous treatments, intended for mathematicians, or are intended for non-mathematicians, and avoid the finer points of the theory. This book bridges the gap between the two types. The text is self-contained in that it includes examples of the use of the various theorems, and any mathematical concepts not usually included in degree courses in physical sciences and engineering are explained. This handbook will be of value to postgraduates and research workers in the physical sciences and in engineering subjects, particularly communications and electronic engineering.

"...the gap between engineering and the mathematical literature and is highly recommended." Choice

Prefacep. xi
Introductionp. 1
Lebesgue integrationp. 4
Introductionp. 4
Riemann integrationp. 4
Null setsp. 5
The Lebesgue integralp. 6
Nomenclaturep. 8
Conditions for integrability; measurabilityp. 9
Functions in L[[superscript p]]p. 12
Integrals in several dimensionsp. 13
Alternative approachesp. 14
Some useful theoremsp. 15
The Minkowski inequalityp. 15
Holder's theoremp. 16
Young's theoremp. 17
The Fubini and Tonelli theoremsp. 18
Two theorems of Lebesguep. 19
Absolute and uniform continuityp. 20
The Riemann--Lebesgue theoremp. 23
Convergence of sequences of functionsp. 24
Introductionp. 24
Pointwise convergencep. 24
Bounded, dominated and monotone convergencep. 25
Uniform convergencep. 26
Convergence in the meanp. 27
Cauchy sequencesp. 29
Local averages and convolution kernelsp. 30
Introductionp. 30
Lebesgue pointsp. 31
Approximate convolution identitiesp. 33
The Dirichlet kernel and Dirichlet pointsp. 35
The functions of du Bois-Reymond and of Fejerp. 36
Carleson's theoremp. 37
Kolmogoroff's theoremp. 37
The Dirichlet conditionsp. 38
Jordan's theoremp. 39
Dini's theoremp. 41
The de la Vallee-Poussin testp. 43
Some general remarks on Fourier transformationp. 44
Introductionp. 44
The definition of the Fourier transformp. 44
Sufficient conditions for transformabilityp. 47
Necessary conditions for transformabilityp. 49
Fourier theorems for good functionsp. 51
Introductionp. 51
Inversion, differentiation and convolution theoremsp. 52
Good functions of bounded supportp. 55
Fourier theorems in L[[superscript p]]p. 60
Basic theorems and definitionsp. 60
More inversion theorems in L[[superscript p]]p. 63
Convolution and product theorems in L[[superscript p]]p. 71
Uncertainty principle and bandwidth theoremp. 75
The sampling theoremp. 77
Hilbert transforms and causal functionsp. 78
Fourier theorems for functions outside L[[superscript p]]p. 81
Introductionp. 81
Functions in class Kp. 82
Convolutions and products in Kp. 85
Functions outside Kp. 87
Miscellaneous theoremsp. 91
Differentiation and integrationp. 91
The Gibbs phenomenonp. 93
Complex Fourier transformsp. 95
Positive-definite and distribution functionsp. 99
Power spectra and Wiener's theoremsp. 102
Introductionp. 102
The autocorrelation functionp. 104
The spectrum and spectral densityp. 106
Discrete spectrap. 109
Continuous spectrap. 113
Miscellaneous theoremsp. 114
Generalized functionsp. 118
Introductionp. 118
The definition of functionals in S'p. 119
Basic theoremsp. 123
Examples of generalized functionsp. 127
Fourier transformation of generalized functions Ip. 135
Definition of the transformp. 135
Simple properties of the transformp. 136
Examples of Fourier transformsp. 137
The convolution and product of functionalsp. 139
Fourier transformation of generalized functions IIp. 145
Functionals of types D' and Z'p. 145
Fourier transformation of functionals in D'p. 149
Transformation of products and convolutions in D'p. 152
Fourier seriesp. 155
Fourier coefficients of a periodic functionp. 155
The convergence of Fourier seriesp. 156
Summability of Fourier seriesp. 158
Mean convergence of Fourier seriesp. 159
Sampling theoremsp. 162
Differentiation and integration of Fourier seriesp. 164
Products and convolutionsp. 166
Generalized Fourier seriesp. 170
Introductionp. 170
Generalized Fourier coefficientsp. 171
The Fourier formulaep. 172
Differentiation, repetition and samplingp. 173
Products and convolutionsp. 175
Bibliographyp. 177
Indexp. 181
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521366885
ISBN-10: 0521366887
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 200
Published: 1st May 1989
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.86 x 15.88  x 1.27
Weight (kg): 0.3