Almost coconvex approximation of continuous periodic functions
If a $2\pi$ -periodic function $f$ continuous on the real axis changes its convexity at $2s, s \in N$, points $y_i : \pi \leq y_{2s} < y_{2s-1} < . . . < y_1 < \pi$ , and, for all other $i \in Z$, $y_i$ are periodically defined, then, for every natural $n \geq N_{y_...
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| Datum: | 2019 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2019
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1444 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | If a $2\pi$ -periodic function $f$ continuous on the real axis changes its convexity at $2s, s \in N$, points $y_i : \pi \leq y_{2s} < y_{2s-1} < . . . < y_1 < \pi$ , and, for all other $i \in Z$, $y_i$ are periodically defined, then, for every natural $n \geq N_{y_i}}$, we determine a trigonometric polynomial $P_n$ of order cn such that $P_n$ has the same convexity as $f$ everywhere except, possibly, small neighborhoods of the points $y_i : (y_i \p_i /n, y_i + \pi /n)$, and
$\| f P_n\| \leq c(s) \omega 4(f, \pi /n)$,,
where $N_{y_i}}$ is a constant depending only on $\mathrm{m}\mathrm{i}\mathrm{n}_{i = 1,...,2s}\{ y_i y_{i+1}\} , c$ and $c(s)$ are constants depending only on $s, \omega 4(f, \cdot )$ is the fourth modulus of smoothness of the function $f$, and $\| \cdot \|$ is the max-norm. |
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