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

G. A. Yusupov

Eurasian mathematical journal, Tome 4 (2013) no. 3, pp. 120-126

Résumé

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 $0$, $m,n,r\in\mathbb N$, $\Omega_m(f^{(r)},\tau)$ 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 $t>0$.