D'Alembert's functional equation on groups

Abstract

Given a (not necessarily unitary) character µ : G → (C \ {0}, •) of a group G we find the solutions g : G → C of the following version of d'Alembert's functional equation g(xy) + µ(y)g(xy −1 ) = 2g(x)g(y), x, y ∈ G. ( * ) The classical equation is the case of µ = 1 and G = R. The non-zero solutions of ( * ) are the normalized traces of certain representations of G on C 2 . Davison proved this via his work [20] on the pre-d'Alembert functional equation on monoids.The present paper presents a detailed exposition of these results working directly with d'Alembert's functional equation. In the process we find for any non-abelian solution g of ( * ) the corresponding solutions w : G → C of w(xy) + w(yx) = 2w(x)g(y) + w(y)g(x), x, y ∈ G.( * * )A novel feature is our use of the theory of group representations and their matrix-coefficients which simplifies some arguments and relates the results to harmonic analysis on groups.

Solutions and stability of a generalization of wilson's equation

“…The purpose in the following is to prove that in Theorem 3.6, case (a), the function g satisfies d'Alembert's short functional equation (3.15). We notice here that the solutions of equation (3.15) are obtained by Stetkaer in [27]. for all x, y ∈ G. Then g is a solution of d'Alembert's short functional equation (3.15).…”

Generalized Hyers-Ulam Stability of Trigonometric Functional Equations

“…In [11], Stetkaer obtained the complex valued solutions of d'Alembert's functional Equation (5) for the case when µ is a character of the group S. The non-zero solutions of the Equation (5) are the normalized traces of certain representations of the group S on C 2 Furthermore, in [12] Ebanks and Stetkaer presented some new results on groups regarding the solutions of Wilson's functional Equation (4) with µ = 1. We shall now also refer to Wilson's first generalization of d'Alembert's functional equation:…”

Stability of a generalization of Wilson’s equation

“…respectively to f (xy) + μ(y)f (xy −1 ) = 2f (x)g(y), x,y ∈ G, (1.4) where μ is a character of the group G. A systematic study of (1.3) and (1.4) has been done by Stetkaer [33,34]. The purpose of the present paper is to extend the theory of Hyers-Ulam stability from (1.1) and (1.2) to (1.3) and (1.4).…”

“…Stetkaer [33] characterized the continuous, complex-valued solutions of d'Alembert's functional equation (1.3) in the abelian and non-abelian case, respectively. In both cases any non-zero continuous solution f has the form f = 1 2 tr(ρ), where ρ is a continuous representation of G on C 2 .…”