Remarks on the Boundedness of Poles of Padé-orthogonal and Padé-Faber approximants

Abstract

Given a function F holomorphic on a neighborhood of some compact subset of the complex plane, we prove that if the zeros of the denominators of (n, mn) orthogonal Padé and Padé-Faber approximants remain uniformly bounded on a sequence of indices {(n, mn)} satisfying sup{mn: n Ԑ N} < ∞. then either F is a polynomial or F has a singularity in the complex plane. In this paper, we relax a condition on the indices mn in results from [N. Bosuwan, On the boundedness of poles of generalized Padé approximants, Adv. Differ. Equ 2019 (137) (2019) https://doi.org/10.1186/sl36fi2-01fl-2081-fl].