Lower bounds for Kolmogorov widths of classes of Poisson integrals

Abstract

We expand the ranges of permissible values of $n$ ($n\in \mathbb{N}$) for which Poisson kernels $P_{q,\beta }(t)=\sum _{k=1}^{\infty }{q}^k\cos \left(kt-\frac{\beta \pi }{2}\right)$, $q\in (0,1)$, $\beta \in \mathbb{R}$, satisfy Kushpel's condition $C_{y,2n}$. As a consequence, we obtain exact values for Kolmogorov widths in the space $C$ ($L$) of classes $C_{\beta ,\infty }^q$ ($C_{\beta ,1}^q$) of Poisson integrals generated by kernels $P_{q,\beta }(t)$ in new situations. It is shown that obtained here results we can't obtain by using methods of finding of exact lower bounds for widths suggested by A. Pinkus.

arXiv comment14 pages, in Ukrainian

References in corpus (1)

• Exact values of Kolmogorov widths of classes of Poisson integrals