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 applied argument assume becomes belongs to L belongs to L2 bounded chapter clearly closed coefficients Combining consider const constant contains continuous converges corresponding defined definition denumerable differs dominated equal equivalent established everywhere exists expression fact finite finite range follows formula Fourier series Fourier transform function function f given harmonic analysis Hence holds increasing infinite integral interval known Lebesgue lemma less lim lim lim sup limit linear mean measurable methods Minkowski inequality modulus monotone normal set null set obtain operator particular periodic functions pertaining points polynomial positive possible present proposition prove result sequence square Tauberian theorem tends term theorem theory translation number true uniformly values vanish Wiener write zero