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

Abstract

We prove that the $n$th Chebyshev polynomial $T_{n}$ of a piecewise Dini-smooth Jordan curve $\Gamma$ satisfies \[ \lim_{n\to\infty}\frac{\|T_{n}\|_{\Gamma}}{\mathrm{cap}(\Gamma)^n}=1, \] where $\|\cdot\|_\Gamma$ is the supremum norm over $\Gamma$ and $\mathrm{cap}(\Gamma)$ 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 $\pi$.