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

Abstract

The study of inner and cyclic functions in ℓ 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 ℓ 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 = 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.