The Fourier Integral and Certain of Its Applications
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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absolutely convergent Fourier applied belongs to Lt class of functions complete the proof const constant convergent Fourier series converges absolutely defined definition denumerable normal set denumerable set equivalent established everywhere finite interval finite number finite range follows formula Fourier integral Fourier series Fourier transform function f(x harmonic analysis Hence Iff(x infinite J-oo Kt(x Lambert series Lebesgue integral Lemma 6u Let f(x let us notice let us put lim f lim sup limited total variation linear lira measurable function Minkowski inequality modulus monotone increasing non-negative null set number of dimensions number of f(x obtain periodic functions Plancherel theorem polynomial positive proof of theorem proposition prove real numbers Riemann Riesz-Fischer theorem Schwarz inequality sequence set of points step-function summable Tauberian theorem tends term by term theorem 15 theorem 22 theory true uniformly values vanish Wiener zero