Let $G$ be a compact group. Let $\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation $$\int_{G}f(xtyt^{-1})\,dt+\int_{G}f(xt\sigma(y)t^{-1})\,dt=2g(x)h(y), \quad x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d'Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$.

Keywords:

Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators. Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators.

MSC Classifications:

39B32, 39B42, 22D10, 22D12, 22D15 show english descriptions Equations for complex functions [See also 30D05]
Matrix and operator equations [See also 47Jxx]
Unitary representations of locally compact groups
Other representations of locally compact groups
Group algebras of locally compact groups 39B32 - Equations for complex functions [See also 30D05]
39B42 - Matrix and operator equations [See also 47Jxx]
22D10 - Unitary representations of locally compact groups
22D12 - Other representations of locally compact groups
22D15 - Group algebras of locally compact groups

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