Best approximation rate of constants by simple partial fractions and Chebyshev alternance

Abstract

We consider the problem of interpolation and best uniform approximation of constants c ≠ 0 by simple partial fractions ρ _n of order n on an interval [a, b]. (All functions and numbers considered are real.) For the case in which n > 4|c|(b − a), we prove that the interpolation problem is uniquely solvable, obtain upper and lower bounds, sharp in order n, for the interpolation error on the set of all interpolation points, and show that the poles of the interpolating fraction lie outside the disk with diameter [a, b]. We obtain an analog of Chebyshev’s classical theorem on the minimum deviation of a monic polynomial of degree n from a constant. Namely, we show that, for n > 4|c|(b − a), the best approximation fraction ρ* _n for the constant c on [a, b] is unique and can be characterized by the Chebyshev alternance of n+1 points for the difference ρ* _n − c. For theminimum deviations, we obtain an estimate sharp in order n.