Zeros of optimal polynomial approximants in [formula omitted

Abstract

The study of inner and cyclic functions in ℓAp 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 ℓAp 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 fd 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 fd and discuss their properties. The method involves the Lagrange multiplier method and a resulting dynamical system.