Decay bounds for non-Hermitian matrix functions

The derivation of a-priori decay bounds for the entries of functions of banded matrices is of interest in a variety of applications. While decay bounds for functions of Hermitian banded matrices have been known for some time, the non-Hermitian case is an especially challenging setting. By using Faber polynomial series we explore the bounds obtainable by extending results for Hermitian matrices to banded non-Hermitian (not necessarily diagonalizable) matrices. Several special cases are treated, together with an application to the inexact Krylov approximation of matrix function evaluations. Numerical experiments illustrate the quality of all new bounds.