Some structural properties of two counter-examples to the Baker–Gammel–Wills conjecture

摘要

I improve the counter-example of Lubinsky, and show that the counter-example of Buslaev is also relevant to the original form of the Baker–Gammel–Wills conjecture. I notice that these counter-examples have only a single spurious pole and that a patchwork of just two subsequences of diagonal Padé approximants provides uniform convergence in compact subsets of |z|<1. I find that both counter-examples can be characterized by the observation that they are associated with bounded J-matrices. I prove a number of results for the convergence of diagonal Padé approximants to functions which have bounded J-matrices.