On trigonometric polynomials spanned by characters of unitary representations (Q1207292)

The author, who died young in spring 1992, generalizes trigonometric polynomials \(\sum^ n_{k=1}c_ ke^{ikx}\) on \(\mathbb{R}\) to (1) \(f(x)=\sum^ n_{k=1}c_ k\chi_{U(k)}(x)\) on topological groups \(G\) with finite dimensional, continuous, irreducible and unitary representations \(U(k)\) \((k=1,2,\dots,n)\). Here the \(\chi_{U(k)}\) are characters of \(U(k)\), and \(c_ k\) are complex constants. There exist, of course, more general trigonometric polynomials on \(G\). The author offers characterizations of such ``special trigonometric polynomials''. One of the simpler results is the following. A trigonometric polynomial \(f:G\to\mathbb{C}\) is special iff it is translation invariant, i.e., (2) \(f(xyx^{-1})=f(y)\) for all \(x,y\in G\). Other results concern special trigonometric polynomials in the space of all continuous and almost periodic functions on \(G\), in particular when \(G\) is compact. Further, attention is paid to the situation where (1) and (2) hold almost everywhere.