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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
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