Integral comparisons of nonnegative positive definite functions on locally compact abelian groups

In this paper, we discuss the following general questions. Let $mu, nu$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying that $$int f dnu le C int f dmu$$ whenever $f$ is a continuous nonnegative positive definite function? How the admissible constants $C$ can be characterized and what is the best value? First we discuss the problem in locally compact Abelian groups and then apply the results to the case when $mu, nu$ are the traces of the usual Lebesgue measure over centered and arbitrary intervals, respectively. This special case was earlier investigated by Shapiro, Montgomery, Halu0027asz and Logan. It is a close companion of the more familiar problem of Wiener, as well.