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A generalized Fourier transformation for L1(G)-Modules

Part of: Harmonic analysis in one variable Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Teng-Sun Liu ,
Arnoud C. M. Van Rooij  and
Ju-Kwei Wang
Teng-Sun Liu
Affiliation:
University of MassachusettsAmherst, Massachusetts 01003 U.S.A.
Arnoud C. M. Van Rooij
Affiliation:
Catholic UniversityNijmegen The Netherlands
Ju-Kwei Wang
Affiliation:
Catholic UniversityNijmegen The Netherlands
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Abstract

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Let G be a compact abelian group with dual Ĝ and let K be a Banach L1 (G)-module. We introduce the notion of character convolution transformation of K which reduces to ordinary Fourier or Fourier-Stieltjes transformation when K is one of the spaces Lp(G), M(G). We show that the question of what maps Ĝ → K extend to multipliers of K is a question of asking for descriptions of the character convolution transforms. In this setting some results of Helson-Edward and Schoenberg-Eberlein find generalizations, as do some classical results, including the inversion formula and the Parseval relation. We then apply these results to transformation groups, obtaining a variant of a theorem of Bochner and an extension of a theorem of Ryan.

MSC classification

Secondary: 42A20: Convergence and absolute convergence of Fourier and trigonometric series 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 3 , June 1984 , pp. 365 - 377
Copyright
Copyright © Australian Mathematical Society 1984

References

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