No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation
UDC 517.5 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 appro...
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| Дата: | 2022 |
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| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2022
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7081 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | UDC 517.5
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 $q \ge 3$. |
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| DOI: | 10.37863/umzh.v74i5.7081 |