On the Exact Values of the Best Approximations of Classes of Differentiable Periodic Functions by Splines

V. F. Babenko ; N. V. Parfinovich

Matematičeskie zametki, Tome 87 (2010) no. 5, pp. 669-683

Résumé

We obtain the exact values of the best $L_1$-approximations of classes $W^rF$, $r\in\mathbb N$, of periodic functions whose $r$th derivative belongs to a given rearrangement-invariant set $F$, as well as of classes $W^rH^\omega$ of periodic functions whose $r$th derivative has a given convex (upward) majorant $\omega(t)$ of the modulus of continuity, by subspaces of polynomial splines of order $m\ge r+1$ and of deficiency 1 with nodes at the points $2k\pi/n$ and $2k\pi/n+h$, $n\in\mathbb N$, $k\in\mathbb Z$, $h\in(0,2\pi/n)$. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

Détail

Keywords: best approximation, differentiable periodic function, polynomial spline, Kolmogorov width, modulus of continuity, extremal subspace, Jackson-type inequality, the space $L_1$, Sobolev class $W_p^r$, the space $L_p$, Orlicz space.