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

Abstract

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.

Keywords

• bernstein width
• kolmogorov width
• linear width
• projection width
• fourier sum
• weyl-nagy class
• class of the generalized poisson integrals
• $(\psi,\bar{\beta})$-integral
• asymptotic equality