$K$-SPHERICAL FUNCTIONS ON ABELIAN SEMIGROUPS. Issue 3 (13th June 2017) - ŁUKASIK, RADOSŁAW
Abstract : We present the form of the solutions $f:S→ \mathbb{C}$ of the functional equation $$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D706}\in K}f(x+\unicode[STIX]x1D706y)=|K|f(x)f(y)\quad \text{for }x, y\in S, \end{eqnarray}$$ where $f$ satisfies the condition $f(\sum _{\unicode[STIX]{x1D706}\in K}\unicode[STIX]x1D706x)≠ 0$ for all $x\in S$, $(S, +)$ is an abelian semigroup and $K$ is a subgroup of the automorphism group of $S$ .