2 results
$K$-SPHERICAL FUNCTIONS ON ABELIAN SEMIGROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 96 / Issue 3 / December 2017
- Published online by Cambridge University Press:
- 13 June 2017, pp. 479-486
- Print publication:
- December 2017
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- Article
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- You have access
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We present the form of the solutions
$f:S\rightarrow \mathbb{C}$ of the functional equation where
$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D706}\in K}f(x+\unicode[STIX]{x1D706}y)=|K|f(x)f(y)\quad \text{for }x,y\in S,\end{eqnarray}$$
$f$ 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 
$S$.
ON EXTENSIONS OF THE GENERALISED JENSEN FUNCTIONS ON SEMIGROUPS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 96 / Issue 1 / August 2017
- Published online by Cambridge University Press:
- 13 February 2017, pp. 110-116
- Print publication:
- August 2017
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- Article
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- You have access
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Assume that
$(G,+)$ is a commutative semigroup, 
$\unicode[STIX]{x1D70F}$ is an endomorphism of 
$G$ and an involution, 
$D$ is a nonempty subset of 
$G$ and 
$(H,+)$ is an abelian group uniquely divisible by two. We prove that if 
$D$ is ‘sufficiently large’, then each function 
$g:D\rightarrow H$ satisfying 
$g(x+y)+g(x+\unicode[STIX]{x1D70F}(y))=2g(x)$ for 
$x,y\in D$ with 
$x+y,x+\unicode[STIX]{x1D70F}(y)\in D$ can be extended to a unique solution 
$f:G\rightarrow H$ of the generalised Jensen functional equation 
$f(x+y)+f(x+\unicode[STIX]{x1D70F}(y))=2f(x)$.
