Optimal polynomial approximants and orthogonal polynomials on the unit circle. An electrostatic approach

TL;DR

This paper investigates the zeros of optimal polynomial approximants in the Hardy space, linking them to orthogonal polynomials on the unit circle and providing an electrostatic interpretation of their distribution.

Contribution

It establishes a novel electrostatic interpretation of zeros of optimal polynomial approximants in the Hardy space, connecting them with orthogonal polynomials on the unit circle.

Findings

Electrostatic laws explain zeros' positions for specific examples.

First step towards understanding zeros of OPA via potential theory.

Links between cyclic functions, OPUC, and OPA are clarified.

Abstract

We explore the connection between two seemingly distant fields: the set of cyclic functions $f$ in a Hilbert space of analytic functions over the unit disc $\D$, on the one hand, and the families of orthogonal polynomials for a weight on the unit circle $\T$ (OPUC), on the other. This link is established by so-called Optimal Polynomial Approximants (OPA) to $1/f$, that is, polynomials $p_n$ minimizing the norm of $1-p_nf$, among all polynomials $p_n$ of degree up to a given $n$. Here, we focus on the particular case of the Hardy space, and an electrostatic interpretation of the zeros of those OPA (and thus, of the corresponding OPUC) is studied. We find the electrostatic laws explaining the position of such zeros for a reduced but significant class of examples. This represents the first step towards a research plan proposed over a decade ago to understand zeros of OPA through their potential theoretic properties.