A criterion for the best approximation of constants by simple partial fractions

Abstract

The problem of the best uniform approximation of a real constant c by real-valued simple partial fractions Rn on a closed interval of the real axis is considered. For sufficiently small (in absolute value) c, |c| ≤ cn, it is proved that Rn is a fraction of best approximation if, for the difference Rn − c, there exists a Chebyshev alternance of n + 1 points on a closed interval. A criterion for best approximation in terms of alternance is stated.