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Integrability of Fourier transforms under an ergodic hypothesis

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Gavin Brown
Gavin Brown
Affiliation:
School of Mathematics University of New South Wales P.O. Box 1 Kensington NSW 2033 Australia
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Abstract

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The object is to unify and complement some recent theorems of Hewitt and Ritter on the integrability of Fourier transforms, but the underlying theme is the ancient one that Plancherel's theorem is the “only” integrability constraint on Fourier transforms. The distinguishing feature of the results is that we restrict attention to positive measures (or functions) which satisfy an ergodic condition and whose transforms are positive. (In fact we employ sums of discrete random variables, a technique which seems to have been largely ignored in context.) The general setting is that of locally compact abelian groups but we are chiefly interested in the line or the circle, and it appears that the theorems are new for these classical groups.

Keywords

Fourier-Stieltjes transformsdiscrete random variables.

MSC classification

Secondary: 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 2 , September 1978 , pp. 129 - 153
Copyright
Copyright © Australian Mathematical Society 1978

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