Comparing integrals of nonnegative and positive definite functions with respect to different measures

This work developed from our previous attempt in the extremal problem of comparing integrals of nonnegative and positive definite functions over different intervals, say on $I:=[-1,1]$ and $J:=[-T,T]$. We found that there is a constant $C(T)$, depending only on $T$ and in the linear order, such that the integral on $J$ is at most $C(T)$ times the integral on $I$. After publishing this result, it turned out that the same extremal problem was already studied by $B$. Logan, who has obtained the same estimate some 20 years before us. Oddly enough, the proofs were somewhat different, yet the (complicated) formula for $C(T)$ of his and ours matched. Although we conjecture that the obtained constant is not always optimal, this is still unresolved. What we can discuss now, is a conjecture of ours stating that in principle our approach is optimal. The way we arrive at this is somewhat long and abstract, relying on two major elements, one being a duality type formula, which is inherently real-valued, and the exploitation of positive definiteness, which is inherently complex valued. The necessity of combining these two is one reason for the technically involved treatement. Further, we consider (integrals with respect to) arbitrary measures on arbitrary locally compact Abelian groups, and the handling of the general setup thus needs the theory of LCA groups in general.