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The influence of the shape of functions on the orders of piecewise polynomial and rational approximation. (English. Russian original) Zbl 1116.41020
Sb. Math. 196, No. 5, 623-648 (2005); translation from Mat. Sb. 196, No. 5, 3-30 (2005).
Let $$\Delta_+^s$$ be the set of functions $$x:I\to \mathbb R$$ on a finite (open, half-open or closed) interval $$I$$ such that divided differences $$[x;t_0,\dots,t_s]$$ of order $$s\in\mathbb N$$ for $$x(t)$$ are non-negative for all systems of distinct points $$t_0,\dots,t_s\in I$$. Let $$\Sigma_{r,n}$$ be the set of piecewise polynomial splines $$\sigma_{r,n}$$ of order $$r$$ with $$n-1$$ free knots and $$R_n$$ be the set of rational functions $$\rho_n$$ of order $$n$$ ($$\rho_n=p_n/q_n$$, where $$p_n,q_n$$ are polynomials of degree $$\leq n-1$$). Let $$B_p$$ be the unit ball in $$L_p(I)$$, $$\Delta^s_+B_p=\Delta^s_+\cap B_p$$, $$E(W,V)=\sup_{x\in W}\inf_{y\in V}\| x-y\| _{L_q(I)}$$. Main results of the paper are: If $$r,s\in\mathbb N$$ and $$1\leq q<p\leq \infty$$, then \begin{aligned} C_1n^{-\min(r,s)}&\leq E(\Delta^s_+B_p, \Sigma_{r,n})\leq C_2n^{-\min(r,s)}, \quad n\geq 1,\\ C_3n^{-s}&\leq E(\Delta^s_+B_p, R_n)\leq C_4n^{-s}, \quad n\geq 1. \end{aligned}
##### MSC:
 41A20 Approximation by rational functions 41A30 Approximation by other special function classes
##### Keywords:
divided differences; splines; rational approximation
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