Chebyshev and Faber Polynomials on Curves with Corners and Cusps

In 1903, Faber introduced a family of polynomials that provide an explicit polynomial representation of analytic functions on Jordan domains with analytic boundaries—giving a constructive version of Runge’s theorem for such domains. In 1920, he further discovered that these so-called Faber polynomials satisfy an asymptotic minimality property: among all polynomials with prescribed leading coefficients, they have asymptotically minimal supremum norm on the domain.

The minimality of Faber polynomials was later extended to curves of sufficient smoothness. However, these arguments break down when a corner is present, and the resulting influence on Faber and Chebyshev polynomials is still largely unknown.

In this talk, I will present a new construction of weighted Faber polynomials that restores minimality for curves with corners and even cusps. This enables us to determine the asymptotic behavior of the associated Chebyshev polynomials and to describe the uniform behavior of Faber polynomials in neighborhoods of corners.

Based on joint work with Erwin Miña-Díaz and Aron Wennman