Coconvex approximation of periodic functions

Abstract

The Jackson inequality E _n (f ) ≤ c ω ₃ (f , π / n ) connects the value of the best uniform approximation E _n (f ) of a 2π-periodic function f : R → R by trigonometric polynomials of order ≤ n — 1 with its third modulus of continuity ω ₃ (f, t ).
In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity on [—π, π) only at every point of a fixed finite set consisting of the even number of points are approximated by polynomials coconvex to them.