Impact of the shape of functions on the orders of piecewise polynomial and rational approximation

V. N. Konovalov

Sbornik. Mathematics, Tome 196 (2005) no. 5, pp. 623-648

Résumé

Let $\Delta^s_+$ be the set of functions $x\colon I\to\mathbb R$ on a finite interval $I$ such that the divided differences $[x;t_0,\dots,t_s]$ of order $s\in\mathbb N$ of these functions are non-negative for all systems of $s+1$ distinct points $t_0,\dots,t_s\in I$. Let $\Sigma_{r,n}=\{\sigma_{r,n}\}$ be the set of piecewise polynomial splines $\sigma_{r,n}$ of order $r$ with $n-1$ free knots, and $R_n=\{\rho_n\}$ the set of rational functions $\rho_n=\widehat\pi_n/\check\pi_n$, where $\widehat\pi_n$ and $\check\pi_n$ are polynomials of order $n$. For the classes $\Delta^s_+B_p:=\Delta^s_+\cap B_p$, where $B_p$ is the unit ball in $L_p$, the precise orders $$ E(\Delta^s_+B_p,\Sigma_{r,n})_{L_q} \asymp n^{-{\min\{r,s\}}}\quad \text{and}\quad E(\Delta^s_+B_p,R_n)_{L_q}\asymp n^{-s} $$ of the best approximations in the $L_q$ metrics are found for $1\leqslant q$.