Functional equations and matrix-valued spherical functions

Abstract

We show that any solution of a certain functional equation, generalizing such equations as Wilson's, Levi-Cività's, Gajda's and the sine addition equation, can be expressed in terms of a matrix-valued spherical function. We compute this matrix-valued function for solutions of various functional equations, and use it to solve the equations. (2000). 39B32 and 43A90. Mathematics Subject Classification

On Cauchy‐type functional equations

“…The solutions are described in Theorem 4.2. As a particular case, we obtain the result given by Stetkaer when G is compact and commutative (see [13]). …”

“…The results obtained in this note may be viewed as some generalizations of important works studied in the literature (see [1,2,5,6,7,11,12,13,14,15,19]). Our work unifies many of the results presented in these references.…”

“…(1.6) Equation (1.6) were considered by Stetkaer in several works (see, e.g., [13,14,15]) and it was solved in the particular case when G is compact and commutative (see [13]). Furthermore, on an abelian group G, the bounded solutions of equations like…”

Stability of generalized Cauchy equations

“…For information concerning the solutions of this equation see, for example, [5][6][7][8][9][10] and [12]. Recently, solutions of (1.1), on various non-abelian groups, have been examined in [4].…”

Superstability of functional equations related to spherical functions

“…For K-spherical functions see [1]. Functional equations related to spherical functions have been studied in [2][3][4]. In the case G is a discrete group and K D fid G g then (1) reduces to…”