OMM Centner

Title: Optimal Polynomial Approximants in $L^p$

Abstract: Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this talk, I will introduce the notion of optimal polynomial approximant in the space $L^p$, $1\leq p\leq\infty$. In the first half of the talk, I will focus the discussion on the location of their zeros. As an example, I will show that if $1 p\infty$, $f\in H^p$, and $f(0)\neq 0$, then there exists a disk, centered at the origin, in which all the associated OPAs are zero-free. In the second half of the talk, I will shed light on an orthogonality condition in $L^p$. This allows one to study the zeros of OPAs through the lens of the Hilbert space $L^2$. To inspire further research in the general theory, I will pose some open questions throughout the discussions.