On Lagrange polynomials and the rate of approximation of planar sets by polynomial Julia sets

TL;DR

This paper improves the understanding of how well polynomial Julia sets can approximate planar sets by analyzing the rate of approximation using modified Lagrange interpolation and specific node sequences.

Contribution

It introduces modified Lagrange interpolation polynomials and demonstrates that certain node classes with subexponential Lebesgue constant growth enhance approximation rates.

Findings

Subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth.

Improved approximation rates of planar sets by polynomial Julia sets in Hausdorff and Klimek metrics.

Analysis of properties of point arrays in the complex plane for approximation purposes.

Abstract

We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed by Lindsey and Younsi and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants improves the approximation rate. To this end we investigate properties of some arrays of points in $\mathbb{C}$. In particular we prove subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth on finite unions of quasiconformal arcs. Finally, for some classes of sets we estimate more precisely the rate of approximation by filled-in Julia sets in Hausdorff and Klimek metrics.