On the one sided translates of powers of continuous functions on locally compact groups (Q1934660)

Let \(G\) be a locally compact group. \(C(G)\) denotes the space of continuous functions on \(G\), and \(C_c(G)\) is the subspace of functions with compact support in \(C(G).\) For a function \(f\) defined on \(G\), let \(V(f)\) and \(V_0(f)\) denote the subspaces generated by all right translates of positive integer powers of \(f\) and non-negative integer powers of \(f\), respectively. Then the author proves the following main result. If \(f \in C_c(g)\) is real valued with a unique maximum (or minimum), then \(V(f)\) is dense in \(L^p(G),\) where \(1 \leq p < \infty\). If \(f \in C(g)\) is real valued which takes a given value at only one point, then \(V_0(f)\) is dense in \(C(G)\). A similar result holds for \(G\) replaced by any Riemannian symmetric space \(X\).