No Jackson-Type Estimates for Piecewise q-Monotone, q≥3, Trigonometric Approximation

We say that a function f ∈ C[a, b] is q-monotone, q ≥ 2, if f ∈ C^q-2(a, b), i.e., belongs to the space of functions with (q -2)th continuous derivative in (a, b), and f^(q-2) is convex in this space. Let f be continuous and 2-periodic. Assume that it changes its q-monotonicity finitely many times in. We are interested in estimating the degree of approximation of f by trigonometric polynomials, which are co-q-monotone with this function, namely, trigonometric polynomials that change their q-monotonicity exactly at the points where f does. These Jackson-type estimates are valid for piecewise monotone (q = 1) and piecewise convex (q = 2) approximations. However, we prove, that no estimates of this kind are valid, in general, for the co-q-monotone approximation with q ≥ 3.