Convergence of Julia polynomials

TL;DR

This paper investigates how Julia polynomials approximate conformal mappings, establishing uniform convergence on Smirnov domains and providing convergence rate estimates based on boundary angles.

Contribution

It demonstrates uniform convergence of Julia polynomials on Smirnov domains and derives convergence rate estimates related to boundary geometry.

Findings

Julia polynomials converge uniformly on Smirnov domains

Convergence rates depend on boundary exterior angles

Provides estimates for approximation speed on piecewise analytic boundaries

Abstract

We study the approximation of conformal mappings with the polynomials defined by Keldysh and Lavrentiev from an extremal problem considered by Julia. These polynomials converge uniformly on the closure of any Smirnov domain to the conformal mapping of this domain onto a disk. We prove estimates for the rate of such convergence on domains with piecewise analytic boundaries, expressed through the smallest exterior angles at the boundary.