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

26.05.2021 14:00 - 15:30

Szilárd Gy. Révész (A. Rényi Institute of the Hungarian Academy)

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 a constant C(T), such that the integral on J is at most C(T) times the integral on I. After publishing this result, it turned out that B. Logan obtained the same estimate some 20 years before us. Oddly enough, the proofs were somewhat different, yet the (complicated) upper estimation for C(T) of his and ours matched. We conjecture that the obtained constant is still not sharp, but this is still unresolved. What we can discuss now, is a conjecture of ours stating that in principle our approach is optimal. We arrive at this 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. Combining these two causes some complications. Further, we consider arbitrary measures μ and ν on arbitrary LCA groups. We aim at telling, under what conditions there exist a constant C, such that the integral of a nonnegative and positive definite function f with respect to μ can never exceed C times the integral of f with respect to ν. Then we make further specializations when the Borel measures μ, ν are both either purely atomic or absolutely continuous with respect to the Haar measure. (Joint with Marcell G. Gaál.)



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