Some extensions of the Levi-Civitá functional equation and richly periodic spaces of functions
Let \(G\) be a topological group. It is known [cf. \textit{E. Shulman}, J. Lond. Math. Soc. 54, No.~1, 111--120 (1996; Zbl 0861.39020)] that if a continuous function on \(G\) satisfies the Levi-Civitá functional equation \[ f(gh) = \sum_{j=1}^N u_j (g) v_j (h) \] for all \(g, h\in G\), then it is a matrix element of a finite-dimensional continuous representation of \(G\). The author studies some classes of functional equations using geometric results on orbits for infinite-dimensional continuous representations of groups. Let \(G\) be a connected topological group. Suppose that continuous functions \(a_j, b_j, u_j, v_j\) on \(G\) satisfy the following functional equation \[ \sum_{j=1}^m a_j (g) b_j (hg) = \sum_{j=1}^n u_j(g) v_j (h) \] for all \(g,h \in G\). The author proves that if the functions \(a_1, \dots, a_m\) are linearly independent, then all \(b_j\) are matrix elements of a continuous finite-dimensional representation of \(G\). For \(G=\mathbb{R}^n\), this means that the \(b_j\) are quasi-polynomials.
orbits for infinite-dimensional continuous representations of groups
scientific article; zbMATH DE number 5872775