Asymptotic two-point approximations by rational functions
For every function \(f(s)\) continuous in \(0\leq s\leq \infty\) possessing asymptotic power expansions both for \(s\to +0\) and for \(s\to+\infty\), the author in the paper under review proves the existence of a sequence of rational functions \(r_n(s)\) with positive denominators satisfying the property \(r_n(s)- f(s)= \begin{cases} O(s^n) \quad & \text{for } s\to +0 \\ O(s^{-n}) \quad & \text{for } s\to+ \infty \end{cases}\) for \(n\in \mathbb{N}_0\), converging uniformly in \([0, \infty]\) to \(f(s) \). Such functions can be considered as generalized two-points Padé approximants. Some illustrative examples are given. \{Reviewer's remark: To the list of references it could be convenient to add some closely related works, among others, e.g. \textit{A. Draux} [Polynômes orthogonaux formels -- applications (1983; Zbl 0504.42001)] and \textit{P. Gonzáles-Vera}, \textit{O. Njåstad}, [J. Comput. Appl. Math. 32, No. 1/2, 97-105 (1990; Zbl 0721.30026)]\}.
scientific article; zbMATH DE number 1112234
scientific article