Lower bounds for Kolmogorov widths of classes of convolutions with Neumann kernel

Abstract

We obtain exact lower bounds for Kolmogorov $n$-widths in spaces $C$ and $L$ of classes of convolutions with Neumann kernel $N_{q,\beta }(t)=\sum _{k=1}^{\infty }\frac{q^k}{k}\cos \left(kt-\frac{\beta \pi }{2}\right)$, $q\in (0,1)$, $\beta \in \mathbb{R}$, for all natural $n$ greater some number which depend only on $q$. The obtained estimates coincide with the best uniform approximations by trigonometric polynomials of mentioned classes. It made possible to obtain exact values for widths of these classes.

arXiv comment21 pages, in Ukrainian