On trigonometric polynomials spanned by characters of unitary representations

Abstract

In this paper our attention is focused on functions which are defined on a topological groupG and can be expressed in the form $$f(x) = \sum\limits_{k = 1}^n {c_k \chi _{^{_{U(k)} } } (x), x \in G,} $$ whereχU(k)(k = 1,⋯, n) are characters of finite-dimensional, continuous, irreducible, unitary representationsU(k) ofG andck are complex coefficients. Such functions constitute a subclass of the family of all trigonometric polynomials onG and they are referred to asspecial trigonometric polynomials. Our main goal is to characterize special trigonometric polynomials in the space of almost periodic functions onG and in the space of all trigonometric polynomials onG. For this purpose, we consider a functional equation of Levi-Civita's type which involves the invariant mean for almost periodic functions. In the case where the groupG is compact we obtain a characterization of special trigonometric polynomials in the space\(\mathfrak{L}_1 (G)\) consisting of Haar integrable functions onG.