SUFFICIENT CONDITION FOR THE BEST UNIFORM APPROXIMATION BY SIMPLE PARTIAL FRACTIONS

摘要

Under the assumption that a certain algebraic identity holds for all n ∈ N (it is verified for n ≤ 5), we prove that a real-valued simple partial fraction R~n with n simple poles lying outside the unit disk is a simple partial fraction of degree at most n of the best uniform approximation of a continuous real-valued functions f on [-1,1] provided that for the difference f - R_n there is a Chebyshev alternance of n + 1 points on [-1,1].The result is applied to the problem of approximation of real constants.