Comparison of the best uniform approximations of analytic functions in the disk and on its boundary

Abstract

Denote by C A the set of functions that are analytic in the disk |z| < 1 and continuous on its closure |z| ≤ 1; let ℜ n , n = 0, 1, 2, ..., be the set of rational functions of degree at most n. Denote by R n (f) (R n (f) A ) the best uniform approximation of a function f ∈ C A on the circle |z| = 1 (in the disk |z| ≤ 1) by the set ℜ n . The following equality is proved for any n ≥ 1: sup{R n (f) A /R n (f): f ∈ C A ∖ ℜ n } = 2. We also consider a similar problem of comparing the best approximations of functions in C A by polynomials and trigonometric polynomials.