Kolmogorov Widths and Bilinear Approximations of the Classes of Periodic Functions of One and Many Variables

Abstract

We obtain the exact order estimates for the Kolmogorov widths of the classes $$ {W}_p^g $$ of periodic functions of one variable generated by the integral operators with kernels g(x, y) from the Nikol’skii–Besov classes $$ {B}_{\mathrm{p},\theta}^{\mathrm{r}} $$ . We also study the behaviors of bilinear approximations to the classes $$ {W}_{\mathrm{p},\alpha}^{\mathrm{r}} $$ of periodic multivariate functions with bounded mixed derivative in the spaces Lq1,q2 for some relations between the parameters r1, p, q1, and q2.