For a sign-based variant of Delsarte and Turán problems, extremal functions exist on LCA groups when the allowed sets are boundary-coherent.
Carathéodory, Über den Variabilitätsbereich der Fourier’schen Konstanten v on positiven harmonischen Funktionen , Rendiconti del Circolo Matematico di Palermo, 32 (1911), 193– 217
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On extremal problems of Delsarte type for positive definite functions on LCA groups
For a sign-based variant of Delsarte and Turán problems, extremal functions exist on LCA groups when the allowed sets are boundary-coherent.