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

Abstract

We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2\pi$-periodic functions $\phi$, such that $\Vert \phi {\Vert }_2\le 1$, with fixed generated kernels ${\Psi }_{\bar{\beta }}$, which have Fourier series of the form $\sum _{k=1}^{\infty }\psi (k)\cos (kt-{\beta }_k\pi /2),$ where $\psi (k)\ge 0,$ $\sum {\psi }^2(k)<\infty ,{\beta }_k\in \mathbb{R},$ in the space $C$. It is shown that for rapidly decrising sequences $\psi (k)$ (in particular, if $\underset{k\to \infty }{\lim }\psi (k+1)/\psi (k)=0$) obtained estimates are asymptotic equalities. We establish that asymptotic equalities for widths of this classes are realized by trigonometric Fourier sums.

arXiv comment14 pages