Cyclicity in Dirichlet-type spaces and extremal polynomials

TL;DR

This paper investigates the approximation of functions in Dirichlet-type spaces by optimal polynomials, analyzing their decay rates, zeros, and connections to a generalized Brown-Shields conjecture.

Contribution

It provides constructive methods for finding optimal polynomials, bounds on their decay rates, and explores zeros and conjectural aspects in Dirichlet-type spaces.

Findings

Derived bounds for decay rates of approximation errors

Analyzed zeros of optimal polynomials

Explored a generalized Brown-Shields conjecture

Abstract

For functions $f$ in Dirichlet-type spaces we study how to determine constructively optimal polynomials $p_n$ that minimize $\|p f-1\|_\alpha$ among all polynomials $p$ of degree at most $n$. Then we give upper and lower bounds for the rate of decay of $\|p_{n}f-1\|_{\alpha}$ as $n$ approaches $\infty$. Further, we study a generalization of a weak version of the Brown-Shields conjecture and some computational phenomena about the zeros of optimal polynomials.