This article is cited in 16 papers

Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in $L_2$

M. Sh. Shabozov^a, G. A. Yusupov<sup>b

a</sup> Institute of Mathematics, Academy of Sciences of Republic of Tajikistan, Dushanbe, Tajikistan
^b Tajik National University, Dushanbe, Tajikistan

Abstract: We find the exact values of the $n$-widths for the classes of periodic differentiable functions in $L_2[0,2\pi]$, satisfying the constraint
$$ \int_0^ht\widetilde\Omega_m^{1/m}(f^{(r)};t)\,dt\le\Phi(h), $$
where $h>0$, $m\in\mathbb N$, $r\in\mathbb Z_+$, $\widetilde\Omega_m^{1/m}(f^{(r)};t)$ is the generalized $m$th order continuity modulus of the derivative $f^{(r)}\in L_2[0,2\pi]$, while $\Phi(t)$ is an arbitrary increasing function such that $\Phi(0)=0$.

Keywords: space of square integrable functions, best approximation, extremal characteristic, generalized continuity modulus, width.

UDC: 517.5

Received: 11.01.2011
Revised: 10.05.2011

 English version: Siberian Mathematical Journal, 2011, 52:6, 1124–1136

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