The Fourier integral and certain of its applications

Norbert Wiener9780521358842, 0521358841

The book was written from lectures given at the University of Cambridge and maintains throughout a high level of rigour whilst remaining a highly readable and lucid account. Topics covered include the Planchard theory of the existence of Fourier transforms of a function of L2 and Tauberian theorems. The influence of G. H. Hardy is apparent from the presence of an application of the theory to the prime number theorems of Hadamard and de la Vallee Poussin. Both pure and applied mathematicians will welcome the reissue of this classic work. For this reissue, Professor Kahane’s Foreword briefly describes the genesis of Wiener’s work and its later significance to harmonic analysis and Brownian motion.

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Table of contents :
Title page……Page 1
Title……Page 2
Publisher……Page 3
Press……Page 4
Dedication……Page 5
CONTENTS……Page 7
Preface……Page 9
1 The Nature of Harmonic Analysis……Page 13
2 The Properties of the Lebesgue Integral……Page 16
3 The Riesz-Fischer Theorem……Page 39
4 Developments in Orthogonal Functions……Page 46
5 The Formal Theory of the Fourier Transform……Page 58
6 Hermite Polynomials and Hermite Functions……Page 63
7 The Generating Function of the Hermite Functions……Page 67
8 The Closure of the Hermite Functions……Page 76
9 The Fourier Transform……Page 79
10 Enunciation of the General Tauberian Theorem……Page 84
11 Lemmas Concerning Functions whose Fourier Transforms Vanish for Large Arguments……Page 92
12 Lemmas on Absolutely Convergent Fourier Series……Page 98
13 The Proof of the General Tauberian Theorem……Page 106
14 The Closure of the Translations of a Function of $L_1$……Page 109
15 The Closure of the Translations of a Function of $L_2$……Page 112
16 The Abel-Tauber Theorem page……Page 116
17 The Prime-Number Theorem as a Tauberian Theorem……Page 124
18 The Lambert-Tauber Theorem……Page 131
19 Ikehara’s Theorem……Page 137
20 The Mean Square Modulus of a Function……Page 150
21 The Spectrum of a Function……Page 162
22 The Spectra of Certain Linear Transforms of a Function……Page 176
23 The Monotoneness of the Spectrum……Page 192
24 The Elementary Properties of Almost Periodic Functions……Page 197
25 The Weierstrass and Parseval Theorems for Almost Periodic Functions……Page 208
Bibliography……Page 212