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

Abstract

We consider the problem of best uniform approximation of real continuous functions by simple partial fractions of degree at most on a closed interval 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 of degree whose poles lie outside the disc with diameter , is a simple partial fraction of the best approximation to if and only if the difference admits a Chebyshev alternance of points on . Then 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.