Bounding Zolotarev numbers using Faber rational functions

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.