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Bounded spherical functions for abstract operators

Part of: Nontrigonometric harmonic analysis Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Joel M. Cohen
Joel M. Cohen
Affiliation:
Department of Mathematics, University of Bari70125 Bari, Italy Department of Mathematics, University of MarylandCollege Park, Maryland 20742, U.S.A.
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Abstract

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The aim of this paper is to study the spherical functions associated to an operator. These functions can be thought of abstractly as being eigenfunctions of the operator which can be expressed in terms of the operator. The meaning of these properties will be made precise as will a notion of boundedness. The results are obtained by studying a specific shift operator on the algebra of functionals on the complex polynomial ring. For the class studied, we obtain ellipses of eigenvalues for which there exist bounded spherical functions. As an application of the results, we study radial functions on discrete groups.

MSC classification

Secondary: 43A90: Spherical functions 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 2 , October 1988 , pp. 195 - 203
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Arveson, W., An invitation to C*-algebras, (Springer-Verlag, New York-Heidelberg-Berlin, 1976).CrossRefGoogle Scholar
[2]Bender, E. A., ‘Asymptotic methods in enumeration’, SIAM Review 16 (1974), 485–517.CrossRefGoogle Scholar
[3]Cartwright, D. I. and Soardi, P. M., ‘Harmonic analysis on the free produce of two cyclic groups’, J. Funct. Anal. 65 (1986), 147–171.CrossRefGoogle Scholar
[4]Choquet, G., Lecture on analysis, vol. II (W. A. Benjamin, Inc., New York-Amsterdam, 1969).Google Scholar
[5]Cohen, J. M., ‘Operator norms on free groups’, Boll. Un. Mat. Ital. B (6) 1 (1982), 1055–1065.Google Scholar
[6]Cohen, J. M., ‘Spectral measure and real measure’, preprint, 1985.Google Scholar
[7]Cohen, J. M. and Trenholme, A. R., ‘Orthogonal polynomials with a constant recursion formula and an application to harmonic analysis’, J. Funct. Anal. 59 (1984), 175–184.CrossRefGoogle Scholar
[8]Figà-Talamanca, A. and Picardello, M. A., Harmonic analysis on free groups, (Lecture Notes in Pure and Appl. Math. 87, Marcel Dekker, New York, 1983).Google Scholar
[9]Kuhn, G. and Soardi, P. M., ‘The Plancherel measure for a polygonal graph’, Ann. Mat. Pura Appl. 144 (1983), 393–401.CrossRefGoogle Scholar
[10]Pytlik, T., ‘Radial functions on free groups and a decomposition of the regular representation into irreducible components’, J. Reine Angew. Math. 326 (1981), 124–135.Google Scholar
[11]Steger, T., Harmonic analysis for an anisotropic walk on a homogeneous tree, (Ph.D. Dissertation, Washington University, 1985).Google Scholar