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On the orders of nonlinear approximations for classes of functions of given form. (English. Russian original) Zbl 1083.41017
Math. Notes 78, No. 1, 88-104 (2005); translation from Mat. Zametki 78, No. 1, 98-114 (2005).
For an interval $$I \subset {\mathbb R}$$ and $$s \in {\mathbb N}$$, the set $$\Delta_+^s B_q$$ is defined as the collection of functions $$f \in L_q(I)$$, $$\| f\| _q \leq 1$$, for which the divided differences $$[f; t_0, \ldots ,t_s]$$ are nonnegative for every $$s+1$$ points $$t_0, \ldots ,t_s \in I$$. Given two subsets, $$W$$ and $$G$$, of $$L_q$$, let $E(W,G)_q=\sup_{f \in W}\, \inf_{g \in G} \| f-g\| _q .$ In the paper under review, $$G$$ is either the set $$R_n$$ of rational functions $$g=u/v$$, where $$u,v$$ are polynomials of order $$\leq n$$, or the set $$\Sigma_{r,n}$$ of splines of order $$r$$ with $$n-1$$ free knots. It is proved that $$E(\Delta_+^s B_\infty, G)_\infty \asymp n^{-1}$$ for both $$G=R_n$$ and $$G=\Sigma_{r,n}$$ if $$s>1$$, $$r=1,2, \ldots$$ (the case $$s=2$$ has been established earlier).
This estimate remains true if $$s=1$$, but only for $$G=\Sigma_{r,n}$$, whereas $$E(\Delta_+^s B_\infty, R_n)$$ does not tend to zero as $$n \to \infty$$. In the case of $$W=\Delta_+^s B_q$$ with $$q< \infty$$, $$E(W,G)_q$$ tends to zero for neither $$G=R_n$$ nor $$G=\Sigma_{r,n}$$.
To compare the above results, the author uses the concept of the so-called pseudodimension of a set of functions. The pseudodimension of $$R_n$$ and $$\Sigma_{r,n}$$ is $$\asymp n$$, and these sets are asymptotically optimal in the sense that $$E(\Delta_+^s B_\infty, G)_\infty \geq c n^{-1}$$, with $$c$$ independent of $$n$$, for any set $$G$$ of pseudodimension $$n$$.
##### MSC:
 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 41A15 Spline approximation 41A20 Approximation by rational functions
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