Asymptotic estimates for the widths of classes of functions of high smothness - Serdyuk, A.S.; Sokolenko, I.V. (Carpathian Mathematical Publications, 2023)
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 $Ψ_{\bar{β}}$, which have Fourier series of the form $$\sum\limits_{k=1}^∞ ψ(k)\cos(kt-β_kπ/2),$$ where $ψ(k)\ge0,$ $\sumψ^2(k)<∞, β_k\in\mathbb{R}$. It is shown that for rapidly decreasing sequences $ψ(k)$ (in particular, if $\lim\limits_{k→∞}{ψ(k+1)}/{ψ(k)}=0$) the obtained estimates are asymptotic equalities. We establish that asymptotic equalities for widths of this classes are realized by trigonometric Fourier sums.