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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

[1]Gulick, S. L., Liu, T. S. and van Rooij, A. C. M., ‘Group algebra modules, II,’ Canad. J. Math. 19 (1967), 151–173.CrossRefGoogle Scholar
[2]Gulick, S. L., Liu, T. S. and van Rooij, A. C. M., ‘Group algebra modules, III,’ Trans. Amer. Math. Soc. 152 (1970), 561–579.CrossRefGoogle Scholar
[3]Gulick, S. L., Liu, T. S. and van Rooij, A. C. M., ‘Group algebra modules, IV,’ Trans. Amer. Math. Soc. 152 (1970), 581–596.Google Scholar
[4]Hewitt, E. and Ross, K. A., Abstract harmonic analysis I–II (Springer Verlag, Berlin, 1963, 1970).Google Scholar
[5]Larsen, R., The multiplier problem (Lecture Notes in Mathematics, 105, Springer Verlag, Berlin, 1969).CrossRefGoogle Scholar
[6]Liu, T.-S., van Rooij, A. C. M. and Wang, J.-K., ‘Group representations in Banach spaces: orbits and almost-penodicity,’ Studies and essays presented to Yu- Why Chen on his 60th Birthday April 1, 1970, pp. 243–254 (Mathematical Research Center, National Taiwan University, Taipei, Taiwan, China, 1970).Google Scholar
[7]Rudin, W., Fourier analysis on groups (Interscience, New York, 1962).Google Scholar
[8]Ryan, R., ‘Fourier transforms of certain classes of integrable functions,’ Trans. Amer. Math. Soc. 105 (1962), 102–111.CrossRefGoogle Scholar
[9]Wang, J. K., ‘Multipliers of commutative Banach algebras,’ Pacific J. Math. 11 (1961), 1131–1149.CrossRefGoogle Scholar