An Introduction to Fourier Analysis and Generalised Functions

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Cambridge University Press, 1958 - Mathematics - 79 pages
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This monograph on generalised functions, Fourier integrals and Fourier series is intended for readers who, while accepting that a theory where each point is proved is better than one based on conjecture, nevertheless seek a treatment as elementary and free from complications as possible. Little detailed knowledge of particular mathematical techniques is required; the book is suitable for advanced university students, and can be used as the basis of a short undergraduate lecture course. A valuable and original feature of the book is the use of generalised-function theory to derive a simple, widely applicable method of obtaining asymptotic expressions for Fourier transforms and Fourier coefficients.
 

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Contents

Introduction
1
12 Knowledge assumed of the reader
2
introductory remarks
3
introductory remarks
8
introductory remarks
10
The theory of generalised functions and their Fourier transforms
15
22 Generalised functions The delta function and its derivatives
16
23 Ordinary functions as generalised functions
21
33 Integral powers
35
34 Integral powers multiplied by logarithms
40
35 Summary of Fourier transform results
42
The asymptotic estimation of Fourier transforms
46
42 Generalisations of the RiemannLebesgue lemma
47
43 The asymptotic expression for the Fourier transform of a function with a finite number of singularities
51
Fourier series
58
52 Determination of the coefficients in a trigonometrical series
60

24 Equality of a generalised function and an ordinary function in an interval
25
25 Even and odd generalised functions
26
26 Limits of generalised functions
27
Definitions properties and Fourier transforms of particular generalised functions
30
32 Nonintegral powers multiplied by logarithms
34
53 Existence of Fourierseries representation for any periodic generalised function
62
54 Examples Poissons summation formula
67
55 Asymptotic behaviour of the coefficients in a Fourier series
71
INDEX
77
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