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29 March 2024
 
  » arxiv » 1212.3364

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Exact values of Kolmogorov widths of classes of Poisson integrals
A. S. Serdyuk ; V. V. Bodenchuk ;
Date 14 Dec 2012
AbstractWe prove that the Poisson kernel $P_{q,eta}(t)=sumlimits_{k=1}^{infty}q^kcos(kt-dfrac{etapi}{2})$, ${qin(0,1)}$, $etainmathbb{R}$ satisfies introduced by Kyshpel’ condition $C_{y,2n}$ begining from some number $n_q$ that depends only on $q$. As a consequence, the lower bounds for Kolmogorov widths in the space $C$ of classes $C_{eta,infty}^q$ of Poisson integrals of functions from unit ball in space $L_infty$ are found for all $ngeqslant n_q$. These estimates coincide with the best uniform approximation of mentioned classes by trigonometric polynomials. As a result, it is found the exact values of the widths of classes $C_{eta,infty}^q$ and shown that the subspaces of trigonometric polynomials of order $n-1$ are optimal for the widths of dimension $2n-1$.
Source arXiv, 1212.3364
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