Optimal polynomial approximants in Lᵖ

Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces, with numerous papers devoted to the properties of their zeros. This talk introduces the notion of optimal polynomial approximant in the space \(L^p\), \(1 \leq p \leq \infty\). The first half focuses on the location of their zeros — for example, 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. The second half sheds light on an orthogonality condition in \(L^p\) that allows one to study the zeros of OPAs through the lens of the Hilbert space \(L^2\). Open questions are posed throughout to inspire further research.