Introduction to Fourier analysis and generalized functions

Title......Page 1
CONTENTS......Page 5
1.1 Scope and purpose of the book......Page 7
1.2 Knowledge assumed of the reader......Page 8
1.3 Fourier series: introductory remarks......Page 9
1.4 Fourier transforms: introductory remarks......Page 14
1.5 Generalised functions: introductory remarks......Page 16
2.1 Good functions and fairly good functions......Page 21
2.2 Generalised functions. The delta function and its derivatives......Page 22
2.3 Ordinary functions as generalised functions......Page 27
2.4 Equality of a generalised function and an ordinary function in an interval......Page 31
2.5 Even and odd generalised functions......Page 32
2.6 Limits of generalised functions......Page 33
3.1 Non-integral powers......Page 36
3.2 Non-integral powers multiplied by logarithms......Page 40
3.3 Integral powers......Page 41
3.4 Integral powers multiplied by logarithms......Page 46
3.5 Summary of Fourier transform results......Page 48
4.1 The Riemann-Lebesgue lemma......Page 52
4.2 Generalisations of the Riemann-Lebesgue lemma......Page 53
4.3 The asymptotic expression for the Fourier transform of a function with a finite number of singularities......Page 57
5.1 Convergence and uniqueness of trigonometrical series as series of generalised functions page......Page 64
5.2 Determination of the coefficients in a trigonometrical series......Page 66
5.3 Existence of Fourier-series representation for any periodic generalised function......Page 68
5.4 Examples. Poisson's summation formula......Page 73
5.5 Asymptotic behaviour of the coefficients in a Fourier series......Page 77
Index......Page 83