Negative results in coconvex approximation of periodic functions

Abstract

We prove, that for each r ∈ N, n ∈ N and s ∈ N there are a collection {yi} 2s i=1 of points y2s < y2s−1 < • • • < y1 < y2s + 2π =: y0 and a 2π -periodic function f ∈ C (∞) (R), such thatand for each trigonometric polynomial Tn of degree ≤ n (of order ≤ 2n + 1), satisfying(2)holds, where cr > 0 is a constant, depending only on r. Moreover, we prove, that for each r = 0, 1, 2 and any such collection {yi} 2s i=1 there is a 2π -periodic function f ∈ C (r) (R), such that (−1) i−1 f is convex on [yi, yi−1], 1 ≤ i ≤ 2s, and, for each sequence {Tn} ∞ n=0 of trigonometric polynomials Tn, satisfying (2), we have lim supω4(f (r) , 1/n) = +∞, where ω4 is the fourth modulus of continuity.