The Fourier Integral and Certain of Its ApplicationsThe 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. |
Contents
CHAPTER I | 46 |
The Closure of the Hermite Functions | 64 |
CHAPTER II | 72 |
SPECIAL TAUBERIAN THEOREMS | 104 |
CHAPTER IV | 150 |
| 200 | |
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Common terms and phrases
absolutely convergent absolutely convergent Fourier argument belongs to L1 coefficients complete the proof const converges absolutely defined denumerable set e-ius e-iux dx eiux equivalent established everywhere exists finite interval finite number finite range fn(x follows Fourier integral Fourier series Fourier transform function f function of L₂ harmonic analysis Hence infinite K₁ L₁ Lambert series Lebesgue integral lemma Let f let us notice let us put lim f(x lim lim lim sup limited total variation M₁ measurable function Minkowski inequality modulus non-negative null set number of f(x periodic functions pertaining Plancherel theorem polynomial proof of theorem proposition prove real numbers Riesz-Fischer theorem Schwarz inequality sequence sinē step-function summable Tauberian theorem theorem 15 theory translation number uniformly values vanish Wiener x)dx zero ίξ λα λη ξε ΣΛ ψη