Solution of a functional equation on compact groups using Fourier analysis

Abdellatif Chahbi, Brahim Fadli, Samir Kabbaj

Abstract


Let \(G\) be a compact group, let \(n \in N\setminus \{0,1\}\) 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 \[ f(xy)+\sum_{k=1}^{n-1}f(\sigma^k(y)x)=nf(x)f(y),\ x,y \in G,\] in terms of unitary characters of \(G\).

Keywords


Functional equation; non-abelian Fourier transform; representation of a compact group.

Full Text:

PDF

References


Akkouchi, M., Bouikhalene, B., Elqorachi, E., Functional equations and K-spherical functions, Georgian Math. J. 15 (2008), 1-20.

An, J., Yang, D.,Nonabelian harmonic analysis and functional equations on compact groups, J. Lie Theory 21 (2011), 427-455.

Badora, R., On a joint generalization of Cauchy's and d'Alembert's functional equations, Aequationes Math. 43 (1992), 72-89.

Chahbi, A., Fadli, B., Kabbaj, S., Functional equations of Cauchy's and d'Alembert's type on compact groups, Proyecciones (Antofagasta) 34 (2015), 297-305.

Chojnacki, W., On some functional equation generalizing Cauchy's and d'Alembert's functional equations, Colloq. Math. 55 (1988), 169-178.

Chojnacki, W., On group decompositions of bounded cosine sequences, Studia Math. 181 (2007), 61-85.

Chojnacki, W., On uniformly bounded spherical functions in Hilbert space, Aequationes Math. 81 (2011), 135-154.

Fadli, B., Zeglami, D., Kabbaj, S., A variant of Wilson's functional equation, Publ. Math. Debrecen, to appear.

Davison, T. M. K., D'Alembert's functional equation on topological groups, Aequationes Math. 76 (2008), 33-53.

Davison, T. M. K., D'Alembert's functional equation on topological monoids, Publ. Math. Debrecen, 75 (2009), 41-66.

de Place Friis, P., D'Alembert's and Wilson's equation on Lie groups, Aequationes Math. 67 (2004), 12-25.

Elqorachi, E., Akkouchi, M., Bakali, A., Bouikhalene, B., Badora's equation on non-Abelian locally compact groups, Georgian Math. J. 11 (2004), 449-466.

Folland, G., A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, FL, 1995.

Shin'ya, H., Spherical matrix functions and Banach representability for locally compact motion groups, Japan. J. Math. (N.S.), 28 (2002), 163-201.

Stetkaer, H., D'Alembert's functional equations on metabelian groups, Aequationes Math. 59 (2000), 306-320.

Stetkaer, H., D'Alembert's and Wilson's functional equations on step 2 nilpotent groups, Aequationes Math. 67 (2004), 241-262.

Stetkaer, H., Properties of d'Alembert functions, Aequationes Math. 77 (2009), 281-301.

Stetkaer, H., Functional Equations on Groups, World Scientfic, Singapore, 2013.

Stetkaer, H., D'Alembert's functional equation on groups, Banach Center Publ. 99 (2013), 173-191.

Stetkaer, H., A variant of d'Alembert's functional equation, Aequationes Math. 89 (2015), 657-662.

Yang, D., Factorization of cosine functions on compact connected groups, Math. Z. 254 (2006), 655-674.

Yang, D., Functional equations and Fourier analysis, Canad. Math. Bull. 56 (2013), 218-224.




DOI: http://dx.doi.org/10.17951/a.2015.69.2.9-15
Date of publication: 2015-12-30 22:51:59
Date of submission: 2015-12-29 21:48:16


Statistics


Total abstract view - 790
Downloads (from 2020-06-17) - PDF - 458

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2015 Abdellatif Chahbi, Brahim Fadli, Samir Kabbaj