Solution of a functional equation on compact groups using Fourier analysis
Abdellatif Chahbi ; Brahim Fadli ; Samir Kabbaj
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 69 (2015), / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

Let G be a compact group, let n∈N∖{0,1} be a fixed element and let σ be a continuous automorphism on G such that σn=I. Using the non-abelian Fourier transform, we determine the non-zero continuous solutions f:G→C of the functional equation f(xy)+∑k=1n-1f(σk(y)x)=nf(x)f(y),x,y∈G, in terms of unitary characters of G.

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:289737 
@article{bwmeta1.element.ojs-doi-10_17951_a_2015_69_2_9-15,
     author = {Abdellatif Chahbi and Brahim Fadli and Samir Kabbaj},
     title = {Solution of a functional equation on compact groups using Fourier analysis},
     journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
     volume = {69},
     year = {2015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2015_69_2_9-15}
}

Abdellatif Chahbi; Brahim Fadli; Samir Kabbaj. Solution of a functional equation on compact groups using Fourier analysis. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 69 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2015_69_2_9-15/

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.