Henrik Stetkær

Functional equations on abelian groups with involution

SummaryWe produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = ΣI2 =1gl(x)hl(y),x, y∈G, where the functionsf,g1,h1 to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar functional equations for a general involution σ.

On the trigonometric subtraction and addition formulas

Summary. We solve extensions to groups of the trigonometric addition and subtraction formulas, in which the plus/minus has been replaced by an involutive group automorphism. The groups need not be abelian.

On a signed cosine equation ofN summands

We find the set of continuous solutionsf, g of the functional equation

Scalar irreducibility of certain eigenspace representations

\begin{array}{*{20}c} {\sum\limits_{n = 0}^{N - 1} {\omega ^n f(x + \omega ^n y) = Ng(x)f(y),} } & {x,y \in C,} \\ \end{array}

Scalar irreducibility of eigenspaces on the tangent space of a reductive symmetric space

(1) whereω = exp(2πi/N) andN ∈ N. We show that if (f, g) ≠ (0, 0) is a continuous solution of (1) theng satisfies the generalized cosine equation

D'Alembert's equation and spherical functions

\begin{array}{*{20}c} {\sum\limits_{n = 0}^{N - 1} {g(x + \omega ^n y) = Ng(x)g(y),} } & {x,y \in C.} \\ \end{array}