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Dalhousie-AARMS AAMP Seminar: Jean-Pierre Garbardo (McMaster University)

October 30, 2020 @ 4:00 pm - 5:00 pm

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

Abstract: If G is a finite abelian group, we call a subset S\subset G symmetric if 0\in G and -x\in S whenever x\in S. We also let S^*=(G\setminus S)\cup\{0\}. We consider the problem of expressing an arbitrary positive definite function F on G as the convolution product of two positive definite functions, one supported on S and the other one supported on S^*. We show that, in the particular case where F is the constant function 1, 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 S\subset G, find the maximum value of the sum \sum_{x\in G}\,f(x) if f(0)=1 and f is a positive definite function on G supported on S. We introduce the notion of  {\it dual Tur\’an problem for S}, which is essentially the Tur\’an problem for the set S^*, and show how the Tur\’an problem for S 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 \mathbb{R}^d.

The Dalhousie-AARMS Analysis-Applied Math-Physics Seminar takes place on Fridays from 4 – 5 pm Atlantic Time over Zoom.  If you would like to attend, please email the organizers for connection details.


October 30, 2020
4:00 pm - 5:00 pm
Event Category:


Zoom seminar


Suresh Eswarathasan