Asymptotics of $L^r$ extremal polynomials for ${0<r\leq\infty}$ on $C^{1+}$ Jordan regions

TL;DR

This paper analyzes the asymptotic behavior of $L^r$-extremal polynomials on $C^{1+}$ Jordan regions, deriving new results for weighted Chebyshev and residual polynomials and connecting them to the Ahlfors problem.

Contribution

It introduces new asymptotic results for $L^r$-extremal polynomials on smooth Jordan regions and relates these to weighted Chebyshev and residual polynomials, extending previous work.

Findings

Asymptotics for $r=2$ are established.

Derived asymptotics for weighted Chebyshev and residual polynomials.

Connected extremal polynomial asymptotics to the Ahlfors problem.

Abstract

We study strong asymptotics of $L^r$-extremal polynomials for measures supported on Jordan regions with $C^{1+}$ boundary for $0<r<\infty$. Using the results for $r=2$, we derive asymptotics of weighted Chebyshev and residual polynomials for upper-semicontinuous weights supported on a $C^{1+}$ Jordan region corresponding to $r=\infty$. As an application, we show how strong asymptotics for extremal polynomials in the Ahlfors problem on a $C^{1+}$ Jordan region can be obtained from that for the weighted residual polynomials. Based on the results we pose a conjecture for asymptotics of weighted Chebyshev and residual polynomials for a $C^{1+}$ arc.