The divergence problem for multipoint Padé approximants of meromorphic functions (Q1096106)

The Taylor series expansion of an analytic function converges inside the largest disk of analyticity, and diverges outside. In 1902 this result was generalized to approximating meromorphic functions with exactly n poles in a disk \(| z| <\rho\) by Padé approximates of type \({\mathbb{R}}^ m_ n\) where the limit is taken as \(m\to \infty\). If in addition the Padé approximates satisfy a type of interpolation requirement then a satisfactory convergence result has been known since 1972 [\textit{E. B. Saff}, J. Approximation Theory 6, 63-67 (1972; Zbl 0241.30013)]. This paper manages to adapt a method of \textit{T. Kakehashi} [Proc. Japan Acad. 32, 707-718 (1956; Zbl 0073.061)] to give a divergence result.