Bounding Zolotarev numbers using Faber rational functions

TL;DR

This paper introduces a method using Faber rational functions to derive tight bounds on Zolotarev numbers, which are then applied to bound singular values of certain structured matrices.

Contribution

It presents a novel construction of Faber rational functions based on Ganelius's approach, providing explicit bounds on Zolotarev numbers and matrix singular values.

Findings

Derived tight bounds on Zolotarev numbers.

Bounded singular values of Cauchy and Vandermonde matrices.

Constructed Faber rational functions using conformal maps.

Abstract

By closely following a construction by Ganelius, we construct Faber rational functions that allow us to derive tight and explicit bounds on Zolotarev numbers. We use our results to bound the singular values of matrices, including complex-valued Cauchy matrices and Vandermonde matrices with nodes inside the unit disk. We construct Faber rational functions using doubly-connected conformal maps and use their zeros and poles to supply shift parameters in the alternating direction implicit method.