TY - JOUR
AU - D. Leviatan
AU - O. V. Motorna
AU - I. A. Shevchuk
PY - 2022/06/17
Y2 - 2024/10/08
TI - No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation
JF - Ukrainsâ€™kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 74
IS - 5
SE - Research articles
DO - 10.37863/umzh.v74i5.7081
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/7081
AB - UDC 517.5We 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 $q \ge 3$.
ER -