Title: Factorization of positive definite functions through convolution and the Turàn problem

Abstract: If is a finite abelian group, we call a subset symmetric if and whenever . We also let . We consider the problem of expressing an arbitrary positive definite function on as the convolution product of two positive definite functions, one supported on and the other one supported on . We show that, in the particular case where is the constant function , this problem is related to the Tur\’an problem for positive definite functions. In the particular case of a finite abelian group, this last problem asks the following question. Given a symmetric set , find the maximum value of the sum if and is a positive definite function on supported on . We introduce the notion of  {\it dual Tur\’an problem for }, which is essentially the Tur\’an problem for the set , and show how the Tur\’an problem for and its dual are related, and how the factorization mentioned above plays a role is solving both those problems. We will then give an overview of how these results can be extended to other abelian groups such as .

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