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On the Fourier transformation of positive, positive definite measures on commutative hypergroups, and dual convolution structures

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Abstract

We show that the support of the Fourier transform of a positive, positive definite measure on a commutative hypergroupK contains a positive character. This generalizes the known fact that the support of the Plancherel measure π contains a positive character (which in general is not the identity character1). It follows that\(supp(\delta _\alpha * \delta _{\bar \alpha } )\) contains a positive character for\(\alpha \in \hat K\) whenever a dual convolution exists. In particular, if1∈supp π, then1 is this character. We also give some further general results about the support of dual convolution products in terms ofsupp π. Some examples associated with Gelfand pairs and, in particular, non-compact Riemannian symmetric spaces of rank 1 are discussed.

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  1. Mathematisches Institut Technische Universität München, Arcisstr. 21, D-8000, München 2, FRG

    Michael Voit

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  1. Michael Voit
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Voit, M. On the Fourier transformation of positive, positive definite measures on commutative hypergroups, and dual convolution structures. Manuscripta Math 72, 141–153 (1991). https://doi.org/10.1007/BF02568271

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  • DOI: https://doi.org/10.1007/BF02568271

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