This article is cited in 1 paper

Best polynomial approximations and widths of certain classes of functions in the space $L_2$

G. A. Yusupov

Tajik National University, 734025, Tajikistan, Dushanbe, Rudaki Av. 17

Abstract: In the paper exact values of the $n$-widths are found for the class of differentiable periodic functions in the space $L_2[0,2\pi]$, satisfying the condition
$$ \left(\int^t_0\tau\Omega^{2/m}_m(f^{(r)},\tau)\,d\tau\right)^{m/2}\le\Phi(t), $$
where $0<t\le\pi/n$, $m,n,r\in\mathbb N$, $\Omega_m(f^{(r)},\tau)$ is the generalized modulus of continuity of order $m$ of the derivative $f^{(r)}\in L_2[0,2\pi]$, and $\Phi(t)$, $0\le t<\infty$ is a continuous non-decreasing function, such that $\Phi(0)=0$ and $\Phi(t)>0$ for $t>0$.

Keywords and phrases: best polynomial approximations, generalized modulus of continuity, extremal characteristics, widths.

MSC: 42A10

Received: 04.10.2011
Revised: 15.06.2012

Language: English