Asymptotic estimates for the widths of classes of functions of high smoothness

A. S. Serdyuk; I. V. Sokolenko

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arXiv · 2304.04586

Asymptotic estimates for the widths of classes of functions of high smoothness

A. S. Serdyuk·I. V. Sokolenko

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Abstract

We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2π$-periodic functions $φ$, such that $\|φ\|_2\le1$, with fixed generated kernels $Ψ_{\barβ}$, which have Fourier series of the form $\sum\limits_{k=1}^\infty ψ(k)\cos(kt-β_kπ/2), $ where $ψ(k)\ge0,$ $\sumψ^2(k)<\infty, β_k\in\mathbb{R},$ in the space $C$. It is shown that for rapidly decrising sequences $ψ(k)$ (in particular, if $\lim\limits_{k\rightarrow\infty}{ψ(k+1)}/{ψ(k)}=0$) 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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BibTeX RIS

A. S. Serdyuk, I. V. Sokolenko. 2023-04-10. Asymptotic estimates for the widths of classes of functions of high smoothness. https://arxiv.org/abs/2304.04586

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