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$K$ -SPHERICAL FUNCTIONS ON ABELIAN SEMIGROUPS

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  13 June 2017

RADOSŁAW ŁUKASIK Open the ORCID record for RADOSŁAW ŁUKASIK [Opens in a new window]
RADOSŁAW ŁUKASIK*
Affiliation:
Institute of Mathematics, University of Silesia, 40-007 Katowice, ul. Bankowa 14, Poland email rlukasik@math.us.edu.pl
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Abstract

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We present the form of the solutions $f:S\rightarrow \mathbb{C}$ of the functional equation

$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D706}\in K}f(x+\unicode[STIX]{x1D706}y)=|K|f(x)f(y)\quad \text{for }x,y\in S,\end{eqnarray}$$
where $f$ satisfies the condition $f(\sum _{\unicode[STIX]{x1D706}\in K}\unicode[STIX]{x1D706}x)\neq 0$ for all $x\in S$ , $(S,+)$ is an abelian semigroup and $K$ is a subgroup of the automorphism group of $S$ .

Keywords

K-spherical functionhomomorphismabelian semigroup

MSC classification

Primary: 39B52: Equations for functions with more general domains and/or ranges
Secondary: 43A90: Spherical functions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 3 , December 2017 , pp. 479 - 486
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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