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
Recommended Citation
Raymond Cheng, William T. Ross, Daniel Seco, "Zeros of optimal polynomial approximants in ℓAp, " Advances in Mathematics, Volume 404, Part A (August, 2022): 1 - 39. DOI: 10.1016/j.aim.2022.108396.