Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-30T03:13:33.194Z Has data issue: false hasContentIssue false
  • English
  • Français

$K$-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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
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. 

References

Badora, R., ‘On the d’Alembert type functional equation in Hilbert algebras’, Funkcial. Ekvac. 43 (2000), 405–418.Google Scholar
Baker, J. A., ‘D’Alembert’s functional equation in Banach algebras’, Acta Sci. Math. (Szeged) 32 (1971), 225–234.Google Scholar
Benson, C., Jenkins, J. and Ratcliff, G., ‘On Gelfand pairs associated with solvable Lie groups’, Trans. Amer. Math. Soc. 321 (1990), 85–116.Google Scholar
Chojnacki, W., ‘On some functional equation generalizing Cauchy’s and d’Alembert’s functional equations’, Colloq. Math. 55 (1988), 169–178.Google Scholar
Corovei, I., ‘The cosine functional equation for nilpotent groups’, Aequationes Math. 15 (1977), 99–106.CrossRefGoogle Scholar
Corovei, I., ‘The d’Alembert functional equation on metabelian groups’, Aequationes Math. 57 (1999), 201–205.Google Scholar
Davison, T. M. K., ‘D’Alembert’s functional equation on topological groups’, Aequationes Math. 76(1–2) (2008), 33–53.CrossRefGoogle Scholar
Davison, T. M. K., ‘D’Alembert’s functional equation on topological monoids’, Publ. Math. Debrecen 75(1–2) (2009), 41–66.Google Scholar
Kannappan, P., ‘The functional equation f (xy) + f (xy -1) = 2f (x)f (y) for groups’, Proc. Amer. Math. Soc. 19 (1968), 69–74.Google Scholar
Łukasik, R., ‘Some generalization of the quadratic and Wilson’s functional equation’, Aequationes Math. 87 (2014), 105–123.Google Scholar
Penney, R. C. and Rukhin, A. L., ‘D’Alembert’s functional equation on groups’, Proc. Amer. Math. Soc. 77 (1979), 73–80.Google Scholar
Shin’ya, H., ‘Spherical matrix functions and Banach representability for locally compact motion groups’, Jpn. J. Math. 28 (2002), 163–201.Google Scholar
Sinopoulos, P., ‘Functional equations on semigroups’, Aequationes Math. 59 (2000), 255–261.Google Scholar
Stetkær, H., ‘D’Alembert’s functional equations on metabelian groups’, Aequationes Math. 56(3) (2000), 306–320.Google Scholar
Stetkær, H., ‘Functional equations and matrix-valued spherical functions’, Aequationes Math. 69 (2005), 271–292.Google Scholar
Stetkær, H., ‘On operator-valued spherical functions’, J. Funct. Anal. 224 (2005), 338–351.Google Scholar
Stetkær, H., Functional Equations on Groups (World Scientific, Hackensack, NJ, 2013).Google Scholar