Functional Equations and μ-Spherical Functions

Abstract

We will study the properties of solutions 𝑓, {𝑔𝑖}, {ℎ𝑖} ∈ 𝐶𝑏(𝐺) of the functional equation where 𝐺 is a Hausdorff locally compact topological group, 𝐾 a compact subgroup of morphisms of 𝐺, χ a character on 𝐾, and μ a 𝐾-invariant measure on 𝐺. This equation provides a common generalization of many functional equations (D'Alembert's, Badora's, Cauchy's, Gajda's, Stetkaer's, Wilson's equations) on groups. First we obtain the solutions of Badora's equation [Aequationes Math. 43: 72–89, 1992] under the condition that (𝐺,𝐾) is a Gelfand pair. This result completes the one obtained in [Badora, Aequationes Math. 43: 72–89, 1992] and [Elqorachi, Akkouchi, Bakali and Bouikhalene, Georgian Math. J. 11: 449–466, 2004]. Then we point out some of the relations of the general equation to the matrix Badora functional equation and obtain explicit solution formulas of the equation in question for some particular cases. The results presented in this paper may be viewed as a continuation and a generalization of Stetkær's, Badora's, and the authors' works.

D'Alembert's functional equation on groups

“…If χ 1 = χ 2 , then χ 1 and χ 2 are linearly independent (Lemma 3.3), so the coefficients to χ 1 (x) and χ 2 (x) are 0. In particular µ(y) χ1 (y) − χ 2 (y) = 0, which gives the desired formula (7). If χ 1 = χ 2 we divide by χ 1 (x) = χ 2 (x) and get again the desired result.…”

“…Abelian solutions. The formula (7) below generalizes Euler's formula (4) for cos to abelian d'Alembert functions.…”

“…Conversely, any function g of the form (7), where χ is a character, is a non-zero abelian solution of (3).…”

“…The Moroccan school has a number of papers for various kinds of spherical functions: [3], [6], [7], [5], [4], [10], [12], [11], [13], [21], [22], [23], [24], [25]. See also [38].…”