Functional equations of Cauchy’s and d’Alembert’s Type on Compact Groups
DOI:
https://doi.org/10.4067/S0716-09172015000300007Keywords:
Non-abelian Fourier transform, Representation of a compact group.Abstract
Using the non-abelian Fourier transform, we find the central continuous solutions of the functional equation
where G is an arbitrary compact group, 
and σ is a continuous automorphism of G, such that σn = I. We express the solutions in terms of the unitary (group) characters of G.
References
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[15] H. Stetkær, D’Alembert’s and Wilson’s functional equations on step 2 nilpotent groups, Aequationes Math., 67, pp. 241-262, (2004).
[16] H. Stetkær, Properties of d’Alembert functions, Aequationes Math., 77, pp. 281-301, (2009).
[17] H. Stetkær, Functional Equations on Groups. World Scientific Publishing Co. (2013).
[18] H. Stetkær, D’Alembert’s functional equation on groups, Banach Center Publ. 99, pp. 173-191, (2013).
[19] D. Yang, Factorization of cosine functions on compact connected groups, Math. Z., 254, pp. 655-674, (2006).
[20] D. Yang, Functional Equations and Fourier Analysis, Canad. Math. Bull. 56, pp. 218-224, (2013).
[2] J. An and D. Yang, Nonabelian harmonic analysis and functional equations on compact groups. J. Lie Theory, 21, pp. 427-455, (2011).
[3] R. Badora, On a joint generalization of Cauchy’s and d’Alembert’s functional equations, Aequationes Math. 43, pp. 72-89, (1992).
[4] W. Chojnacki, Group representations of bounded cosine functions. J. Reine Angew. Math., 478, pp. 61-84, (1986).
[5] W. Chojnacki, On some functional equation generalizing Cauchy’s and d’Alembert’s functional equations. Colloq. Math., 55, pp. 169-178, (1988).
[6] W. Chojnacki, On group decompositions of bounded cosine sequences. Studia Math., 181, pp. 61-85, (2007).
[7] W. Chojnacki,On uniformly bounded spherical functions in Hilbert space, Aequationes Math., 81, pp. 135-154, (2011).
[8] T.M.K. Davison, D’Alembert’s functional equation on topological groups. Aequationes Math., 76, pp. 33-53, (2008).
[9] T.M.K. Davison, D’Alembert’s functional equation on topological monoids. Publ. Math. Debrecen, 75, pp. 41-66, (2009).
[10] E. Elqorachi, M. Akkouchi, A. Bakali, B. Bouikhalene, Badora’s equation on non-Abelian locally compact groups. Georgian Math. J., 11, pp. 449-466, (2004).
[11] P. de Place Friis, D’Alembert’s and Wilson’s equation on Lie groups. Aequationes Math., 67, pp. 12-25, (2004).
[12] G. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, FL, (1995).
[13] H. Shin’ya, Spherical matrix functions and Banach representability for lo- cally compact motion groups. Japan. J. Math. (N.S.), 28, pp. 163- 201, (2002).
[14] H. Stetkær, D’Alembert’s functional equations on metabelian groups, Aequationes Math., 59, pp. 306-320, (2000).
[15] H. Stetkær, D’Alembert’s and Wilson’s functional equations on step 2 nilpotent groups, Aequationes Math., 67, pp. 241-262, (2004).
[16] H. Stetkær, Properties of d’Alembert functions, Aequationes Math., 77, pp. 281-301, (2009).
[17] H. Stetkær, Functional Equations on Groups. World Scientific Publishing Co. (2013).
[18] H. Stetkær, D’Alembert’s functional equation on groups, Banach Center Publ. 99, pp. 173-191, (2013).
[19] D. Yang, Factorization of cosine functions on compact connected groups, Math. Z., 254, pp. 655-674, (2006).
[20] D. Yang, Functional Equations and Fourier Analysis, Canad. Math. Bull. 56, pp. 218-224, (2013).
How to Cite
[1]
A. Chahbi, B. Fadli, and S. Kabbaj, “Functional equations of Cauchy’s and d’Alembert’s Type on Compact Groups”, Proyecciones (Antofagasta, On line), vol. 34, no. 3, pp. 297-305, 1.
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