Zeros of optimal polynomial approximants in $\ell^p_{A}$
Abstract
The study of inner and cyclic functions in $\ell^p_A$ spaces requires a better understanding of the zeros of the socalled 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 arXiv:
 arXiv:2104.08014
 Bibcode:
 2021arXiv210408014C
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Functional Analysis;
 Primary: 47A15. Secondary: 30C15;
 30H99