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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 ϕ: G × G → 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 ∈ X → ϕ(x − y): y ∈ 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.