A class of integral functional equations with involution

D ZEGLAMI, Samir KABBAJ, B FADLI

Résumé


Let G be a locally compact group, t an involution of G and let m be a regular, compactly supported, complexvalued Borel measure on G. We find all complex-valued continuous solutions of the functional equation Z G f f (xyt)+ f (xt(y)t)gdm(t) = 2g(x) f (y); x;y 2 G; generalizing our result from [17] where this was obtained for t replaced by the group inversion and, when f is central, we determine the continuous solutions f ;g : G!C of the functional equation Z G f f (xyt)+ f (xt(y)t)gdm(t) = 2 f (x)g(y); x;y 2 G; in terms of characters, additive maps and matrix elements of two-dimensional representations of G. To show that these two families of equations provide a joint generalization of an infinite number of functional equations, many applications are presented.

Mots-clés


Kannappan’s functional equation, involution, Wilson, character, additive map, irreducible representation.

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