# Dalhousie-AARMS AAMP Seminar: Jean-Pierre Garbardo (University of Waterloo)

## 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 $S\subset G$ {\it 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.*