A criterion for the best uniform approximation by simple partial fractions in terms of alternance

We consider the problem of best uniform approximation of real continuous functions $f$ by simple partial fractions of degree at most $n$ on a closed interval $S$ of the real axis. We get analogues of the classical polynomial theorems of Chebyshev and de la Vallée-Poussin. We prove that a real-valued simple partial fraction $R_n$ of degree $n$ whose poles lie outside the disc with diameter $S$, is a simple partial fraction of the best approximation to $f$ if and only if the difference $f-R_n$ admits a Chebyshev alternance of $n+1$ points on $S$. Then $R_n$ is the unique fraction of best approximation. We show that the restriction on the poles is unimprovable. Particular cases of the theorems obtained have been stated by various authors only as conjectures.