
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.
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