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Exact values of Kolmogorov widths of classes of Poisson integrals | A. S. Serdyuk
; V. V. Bodenchuk
; | Date: |
14 Dec 2012 | Abstract: | We 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 | Services: | Forum | Review | PDF | Favorites |
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