$K$-SPHERICAL FUNCTIONS ON ABELIAN SEMIGROUPS. Issue 3 (13th June 2017)

• Record Type: Journal Article
• Authors: ŁUKASIK, RADOSŁAW
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 96:Issue 3(2017)
• Page Start: 479
• Page End: 486
• Publication Date: 2017-06-13
• Subjects: primary 39B52, -- secondary 43A90; K-spherical function, -- homomorphism, -- abelian semigroup; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972717000417
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract : 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 $S$ .