Zeros of optimal polynomial approximants in ${\ell }_A^p$

Abstract

The study of inner and cyclic functions in ${\ell }_A^p$ 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 ${\ell }_A^p$ 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 $p\ne 2$. 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.