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

TL;DR

The paper develops a unified framework for the strong asymptotics of $L^r$-extremal polynomials on sets bounded by $C^{1+}$ Jordan curves, spanning $0<r\le \infty$ and general point $z_0$ in the exterior domain. By linking Christoffel functions, Widom factors, and Szegő-type data through conformal maps, outer/Szegő functions, and Faber polynomials, it establishes universal lower/upper bounds and proves sharp asymptotics: $\lim_{n\to\infty} W_{r,n}(\mu,z_0)^r = S(f_{z_0},z_0)$ with strong convergence in Hardy spaces to the extremal kernel function $F_{\mu,z_0,r}$. The results extend Szegő theory to general measures and weights, derive strong Szegő asymptotics for $L^r$ extremals, and apply the framework to the Ahlfors problem, deriving precise asymptotics and highlighting connections to Szegő kernels; the paper also raises a conjecture for $C^{1+}$ arcs. Overall, the work significantly advances understanding of extremal polynomial behavior in complex potential theory and its geometric applications.

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.

Theorems & Definitions (31)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Conjecture 1.7
• Lemma 2.1
• Theorem 2.2
• Definition 2.3