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

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$f^{(q-2)}$isconvexin$(a,b)$\mathrm{.}Let$f$becontinuousand$2\pi$-periodic,andchangeits$q$-monotonicityfinitelymanytimesin$[-\pi,\pi]$\mathrm{.}Weareinterestedinestimatingthedegreeofapproximationof$f$bytrigonometricpolynomialswhichareco-$q$-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 $q\ge 3$.