Norms of Chebyshev and Faber polynomials on curves with corners and cusps

2025/09/26 by Miña-Díaz, Erwin, Rubin, Olof, Wennman, Aron
#30C10 #30E10 #41A10 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.22588

Abstract

We prove that the nth Chebyshev polynomial T_n of a piecewise Dini-smooth Jordan curve Γ satisfies lim_n→∞\frac‖T_n‖_Γcap(Γ)ⁿ=1, where ‖⋅‖_Γ is the supremum norm over Γ and cap(Γ) its logarithmic capacity. This extends earlier results for smooth curves to curves with corner singularities, including cusps. The proof makes use of weighted Faber polynomials, which we analyze using a Fourier analytic representation of the standard Faber polynomials due to Pommerenke. We moreover obtain new asymptotic bounds for the norm of Faber polynomials which are sharp if, for instance, all corners have exterior angle greater than π.

Citations

• Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane
• Computing Chebyshev polynomials using the complex Remez algorithm
• Asymptotics of Chebyshev Polynomials, IV. Comments on the Complex Case
• Asymptotics of Chebyshev Polynomials, III. Sets Saturating SzegŐ, Schiefermayr, and Totik--Widom Bounds
• On Chebyshev polynomials in the complex plane
• Szego\H o-Widom asymptotics of Chebyshev Polynomials on Circular Arcs

Related

• Chebyshev polynomials on equipotential curves
• Chebyshev Polynomials on a System of Continua
• On Chebyshev polynomials in the complex plane
• Chebyshev polynomials in the complex plane and on the real line
• Szego\H o-Widom asymptotics of Chebyshev Polynomials on Circular Arcs