Fundamental Sets of Functions on Locally Compact Abelian Groups

Let G be a locally compact Abelian group, and let X be a compact set of G. Given a positive definite function phi: G x G -> DOUBLE-STRUCK CAPITAL C whose real part is continuous at neutral element of G, we research a necessary and sufficient setting for the linear span of the set {x is an element of X -> phi(x - y): y is an element of X} to be dense in C(X) in the topology of uniform convergence. The context treated that is abstract encompasses classical cases of the literature, while other examples are entirely new.