Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. II
It is shown that the lower bounds of the Kolmogorov widths $d_{2n}$ in the space $C$ established in the first part of our work for the function classes that can be represented in the form of convolutions of the kernels $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \...
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| Datum: | 2015 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2015
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2041 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | It is shown that the lower bounds of the Kolmogorov widths $d_{2n}$ in the space $C$ established in the first part of our work for the function classes that can be represented in the form of convolutions of the kernels
$${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),\kern1em h>0,\kern1em \beta \in \mathbb{R},}$$
with functions $φ ⊥ 1$ from the unit ball in the space $L_{∞}$ coincide (for all $n ≥ nh$) with the best uniform approximations of these classes by trigonometric polynomials whose order does not exceed $n − 1$.
As a result, we obtain the exact values of widths for the indicated classes of convolutions. Moreover, for all $n ≥ nh$, we determine the exact values of the Kolmogorov widths $d_{2n-1}$ in the space $L_1$ of classes of the convolutions of functions $φ ⊥ 1$ from the unit ball in the space $L_1$ with the kernel $H_{h,β}$. |
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