An Analog of the Haar Condition for Simple Partial Fractions

Abstract

We prove that for a continuous real-valued function f on the segment [−1, 1] a real-valued simple partial fraction R n with n distinct poles outside the unit disk is a fraction of degree at most n of best approximation and is unique if and only if for the difference f − R n on [−1, 1] there exists a Chebyshev alternance of n + 1 points. The result is applied to the problem on approximation of real constants.