2019
Том 71
№ 11

All Issues

Bodenchuk V. V.

Articles: 2
Article (Ukrainian)

Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. II

Bodenchuk V. V., Serdyuk A. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1011-1018

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 $H_{h,β}$.

Article (Ukrainian)

Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. I

Bodenchuk V. V., Serdyuk A. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 719-738

We prove that the kernels of analytic functions of the form $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),}h>0,\beta \in \mathbb{R},$$ satisfy Kushpel’s condition $C_{y,2n}$ starting from a certain number $n_h$ explicitly expressed via the parameter $h$ of smoothness of the kernel. As a result, for all $n ≥ n_h$ , we establish lower bounds for the Kolmogorov widths $d_{2n}$ in the space $C$ of functional classes that can be represented in the form of convolutions of the kernel $H_{h,β}$ with functions $φ⊥1$ from the unit ball in the space $L_{∞}$.