@article{Leviatan_Motorna_Shevchuk_2022, title={No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation}, volume={74}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/7081}, DOI={10.37863/umzh.v74i5.7081}, abstractNote={<p>UDC 517.5</p> <p>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 $q \ge 3$.</p>}, number={5}, journal={Ukrainsâ€™kyi Matematychnyi Zhurnal}, author={Leviatan D. and MotornaO. V. and ShevchukI. A.}, year={2022}, month={Jun.}, pages={662 - 675} }