A Criterion for the Best Approximation of Constants by Simple Partial Fractions

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