# Solution of a functional equation on compact groups using Fourier analysis

• Volume: 69, Issue: 2
• ISSN: 0365-1029

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## Abstract

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Let $G$ be a compact group, let $n\in N\setminus \left\{0,1\right\}$ be a fixed element and let $\sigma$ be a continuous automorphism on $G$ such that ${\sigma }^{n}=I$. Using the non-abelian Fourier transform, we determine the non-zero continuous solutions $f:G\to C$ of the functional equation $f\left(xy\right)+\sum _{k=1}^{n-1}f\left({\sigma }^{k}\left(y\right)x\right)=nf\left(x\right)f\left(y\right),\phantom{\rule{4pt}{0ex}}x,y\in G,$ in terms of unitary characters of $G$.

## How to cite

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Abdellatif Chahbi, Brahim Fadli, and Samir Kabbaj. "Solution of a functional equation on compact groups using Fourier analysis." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 69.2 (2015): null. <http://eudml.org/doc/289737>.

@article{AbdellatifChahbi2015,
abstract = {Let $G$ be a compact group, let $n \in N\setminus \lbrace 0,1\rbrace$ be a fixed element and let $\sigma$ be a continuous automorphism on $G$ such that $\sigma ^n=I$. Using the non-abelian Fourier transform, we determine the non-zero continuous solutions $f:G \rightarrow C$ of the functional equation $f(xy)+\sum \_\{k=1\}^\{n-1\}f(\sigma ^k(y)x)=nf(x)f(y),\ x,y \in G,$ in terms of unitary characters of $G$.},
author = {Abdellatif Chahbi, Brahim Fadli, Samir Kabbaj},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Functional equation; non-abelian Fourier transform; representation of a compact group.},
language = {eng},
number = {2},
pages = {null},
title = {Solution of a functional equation on compact groups using Fourier analysis},
url = {http://eudml.org/doc/289737},
volume = {69},
year = {2015},
}

TY - JOUR
AU - Abdellatif Chahbi
AU - Samir Kabbaj
TI - Solution of a functional equation on compact groups using Fourier analysis
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2015
VL - 69
IS - 2
SP - null
AB - Let $G$ be a compact group, let $n \in N\setminus \lbrace 0,1\rbrace$ be a fixed element and let $\sigma$ be a continuous automorphism on $G$ such that $\sigma ^n=I$. Using the non-abelian Fourier transform, we determine the non-zero continuous solutions $f:G \rightarrow C$ of the functional equation $f(xy)+\sum _{k=1}^{n-1}f(\sigma ^k(y)x)=nf(x)f(y),\ x,y \in G,$ in terms of unitary characters of $G$.
LA - eng
KW - Functional equation; non-abelian Fourier transform; representation of a compact group.
UR - http://eudml.org/doc/289737
ER -

## References

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