Exact values of Kolmogorov widths of classes of Poisson integrals

Abstract

We prove that the Poisson kernel $P_{q,\beta }(t)=\sum _{k=1}^{\infty }{q}^k\cos (kt-\frac{\beta \pi }{2})$, $q\in (0,1)$, $\beta \in \mathbb{R}$, satisfies Kushpel's condition $C_{y,2n}$ beginning with a number $n_q$ where $n_q$ is the smallest number $n\ge 9$, for which the following inequality is satisfied: $$\frac{43}{10(1-q)}{q}^{\sqrt{n}}+\frac{160}{57(n-\sqrt{n})}\frac{q}{(1-q{)}^2}\le (\frac{1}{2}+\frac{2q}{(1+q^2)(1-q)})(\frac{1-q}{1+q}{)}^{\frac{4}{1-q^2}}.$$ As a consequence, for all $n\ge n_q$ we obtain lower bounds for Kolmogorov widths in the space $C$ of classes $C_{\beta ,\infty }^q$ of Poisson integrals of functions that belong to the unit ball in the space $L_{\infty }$. The obtained estimates coincide with the best uniform approximations by trigonometric polynomials for these classes. As a result, we obtain exact values for widths of classes $C_{\beta ,\infty }^q$ and show that subspaces of trigonometric polynomials of order $n-1$ are optimal for widths of dimension $2n$.

Cited by in corpus (2)

• sk-Spline interpolation on R^n
• Lower bounds for Kolmogorov widths of classes of Poisson integrals