Zeros of optimal polynomial approximants in ℓp

DOI

10.1016/j.aim.2022.108396

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.

Document Type

Article

Publication Date

8-6-2022

Publisher Statement

Copyright © 2022, ScienceDirect.

DOI: 10.1016/j.aim.2022.108396.

The definitive version is available at: https://www.sciencedirect.com/science/article/pii/S0001870822002122

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