No Jackson-type estimates for piecewise $q$-monotone, $q\ge 3$, trigonometric approximation

Abstract

We say that a function $f\in C[a,b]$ is $q$-monotone, $q\ge 3$, if $f\in C^{q-2}(a,b)$ and $f^{(q-2)}$ is convex in $(a,b)$. Let $f$ be continuous and $2\pi$-periodic, and change its $q$-monotonicity finitely many times in $[-\pi ,\pi ]$. We are interested in estimating the degree of approximation of $f$ by trigonometric polynomials which are co-$q$-monotone with it, namely, trigonometric polynomials that change their $q$-monotonicity exactly at the points where $f$ does. Such Jackson type estimates are valid for piecewise monotone ($q=1$) and piecewise convex (q=2) approximations. However, we prove, that no such estimates are valid, in general, for co-$q$-monotone approximation, when $q\ge 3$.