Olof Rubin: Chebyshev and Faber Polynomials on Curves with Corners and Cusps

Abstract: In 1903, Faber introduced a family of polynomials that provide an explicit series representation for analytic functions on a Jordan domain, thereby extending the classical Taylor expansion beyond the disk. In 1920, he showed that these Faber polynomials have an asymptotic minimality property: among all polynomials with a fixed leading coefficient, they are asymptotically minimal in supremum norm on the domain. The actual minimizers are the Chebyshev polynomials, and Faber’s observation shows that the two sequences become asymptotically comparable. This minimality result was later proved for domains with sufficiently smooth boundaries, but the presence of corners breaks the classical arguments.

I will present a new construction of weighted Faber polynomials that recovers asymptotic minimality for domains with corners and even cusps. This allows us to determine the asymptotic behavior of the associated Chebyshev polynomials. If time permits, I will also describe the uniform behavior of Faber polynomials in neighborhoods of corner singularities.

Joint work with Erwin Miña-Díaz and Aron Wennman.