Sufficient condition for the best uniform approximation by simple partial fractions

Abstract

Under the assumption that a certain algebraic identity holds for all $ n\in \mathbb{N} $ (it is verified for n ≤ 5), we prove that a real-valued simple partial fraction R n with n simple poles lying outside the unit disk is a simple partial fraction of degree at most n of the best uniform approximation of a continuous real-valued functions f on [−1, 1] provided that for the difference f − R n there is a Chebyshev alternance of n + 1 points on [−1, 1]. The result is applied to the problem of approximation of real constants. Bibliography: 8 titles.