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Harmonic Analysis By Henry Helson
English | PDF | 2010 | 238 Pages | ISBN : 9380250053 | 13.96 MB
Harmonic Analysis used to go by the more prosaic name Fourier Series. Its elevation in status may be due to recognition of its crucial place in the intersection of function theory, functional analysis, and real variable theory; or perhaps merely to the greater weightiness of our times. The 1950's were a decade of progress, in which the author was fortunate to be a participant. Some of the results from that time are included here. This book begins at the beginning, and is intended as an introduction for students who have some knowledge of complex variables, measure theory, and linear spaces. Classically the subject is related to complex function theory. We follow that tradition rather than the modern direction, which prefers real methods in order to generalize some of the results to higher dimensional spaces. In this edition there is a full presentation of Bochner's theorem, and a new chapter treats the duality theory for compact and discrete abelian groups. Then the author indulges his own experience and tastes, presenting some of his own theorems, a proof by C. L. Siegel of Minkowski's theorem, applica- tions to probability, and in the last chapter two different methods of proving the theorem of Weyl on equidistribution modulo 1 of (P(k)), where P is a real polynomial with at least one irrational coefficient.
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