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Exact values of Kolmogorov widths of classes of analytic functions | | A. S. Serdyuk
; V. V. Bodenchuk
; | | Date: |
11 Oct 2014 | | Abstract: | We prove that kernels of analytic functions of kind
$H_{h,eta}(t)=sumlimits_{k=1}^{infty}frac{1}{cosh
kh}cosBig(kt-frac{etapi}{2}Big)$, $h>0$, ${etainmathbb{R}}$,
satisfies Kushpel’s condition $C_{y,2n}$ beginning with some number $n_h$ which
is explicitly expressed by parameter $h$ of smoothness of the kernel. As a
consequence, for all $ngeqslant n_h$ we obtain lower bounds for Kolmogorov
widths $d_{2n}$ of functional classes that are representable as convolutions of
kernel $H_{h,eta}$ with functions $varphiperp1$, which belong to the unit
ball in the space $L_{infty}$, in the space $C$. 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 mentioned
classes of convolutions. Also for all $ngeqslant n_h$ we obtain exact values
for Kolmogorov widths $d_{2n-1}$ of classes of convolutions of functions
$varphiperp1$, which belong to the unit ball in the space $L_1$, with kernel
$H_{h,eta}$ in the space $L_1$. | | Source: | arXiv, 1410.2966 | | Services: | Forum | Review | PDF | Favorites | |
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