Zeros of optimal polynomial approximants in $\ell^p_{A}$

TL;DR

This paper characterizes the zeros of optimal polynomial approximants in ll^p_A spaces, showing they lie outside a specific radius depending on p, and develops a method to find extremal functions minimizing root modulus.

Contribution

It determines the zero locations of optimal polynomial approximants in ll^p_A spaces and introduces a systematic method to find extremal functions minimizing root modulus.

Findings

Zeros lie outside a p-dependent radius

Explicit radius values for p

Method to find extremal functions using Lagrange multipliers

Abstract

The study of inner and cyclic functions in $\ell^p_A$ spaces requires a better understanding of the zeros of the so-called optimal polynomial approximants. We determine that a point of the complex plane is the zero of an optimal polynomial approximant for some element of $\ell^p_A$ if and only if it lies outside of a closed disk (centered at the origin) of a particular radius which depends on the value of $p$. We find the value of this radius for $p\neq 2$. In addition, for each positive integer $d$ there is a polynomial $f_d$ of degree at most $d$ that minimizes the modulus of the root of its optimal linear polynomial approximant. We develop a method for finding these extremal functions $f_d$ and discuss their properties. The method involves the Lagrange multiplier method and a resulting dynamical system.