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Mat. Sb., 2005, Volume 196, Number 5, Pages 3–30 (Mi msb1355)  

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

V. N. Konovalov

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: Let $\Delta^s_+$ be the set of functions $x\colon I\to\mathbb R$ on a finite interval $I$ such that the divided differences $[x;t_0,…,t_s]$ of order $s\in\mathbb N$ of these functions are non-negative for all systems of $s+1$ distinct points $t_0,…,t_s\in I$. 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
$$ E(\Delta^s_+B_p,\Sigma_{r,n})_{L_q} \asymp n^{-{\min\{r,s\}}}\quad and\quad E(\Delta^s_+B_p,R_n)_{L_q}\asymp n^{-s} $$
of the best approximations in the $L_q$ metrics are found for $1\leqslant q<p\leqslant\infty$.

DOI: https://doi.org/10.4213/sm1355

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English version:
Sbornik: Mathematics, 2005, 196:5, 623–648

Bibliographic databases:

UDC: 517.5
MSC: 41A15, 41A25
Received: 10.06.2004 and 01.09.2004

Citation: V. N. Konovalov, “Impact of the shape of functions on the orders of piecewise polynomial and rational approximation”, Mat. Sb., 196:5 (2005), 3–30; Sb. Math., 196:5 (2005), 623–648

Citation in format AMSBIB
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