Asymptotic estimates for the widths of classes of functions of high smothness
We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2\pi$-periodic functions $\varphi$, such that $\|\varphi\|_2\le1$, with fixed generated kernels $\Psi_{\bar{\beta}}$, which have Fourier series of the form $$\sum\limits_{k=1}^\inft...
Main Authors: | , |
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Format: | Article |
Language: | English |
Published: |
Vasyl Stefanyk Precarpathian National University
2023-06-01
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Series: | Karpatsʹkì Matematičnì Publìkacìï |
Subjects: | |
Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/6705 |
Summary: | We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2\pi$-periodic functions $\varphi$, such that $\|\varphi\|_2\le1$, with fixed generated kernels $\Psi_{\bar{\beta}}$, which have Fourier series of the form $$\sum\limits_{k=1}^\infty \psi(k)\cos(kt-\beta_k\pi/2),$$ where $\psi(k)\ge0,$ $\sum\psi^2(k)<\infty, \beta_k\in\mathbb{R}$. It is shown that for rapidly decreasing sequences $\psi(k)$ (in particular, if $\lim\limits_{k\rightarrow\infty}{\psi(k+1)}/{\psi(k)}=0$) the obtained estimates are asymptotic equalities. We establish that asymptotic equalities for widths of this classes are realized by trigonometric Fourier sums. |
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ISSN: | 2075-9827 2313-0210 |