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Estimates of Kolmogorov type diameters for classes of differentiable periodic functions. (Russian) Zbl 0561.41022

Let X be a linear vector space with the norm \(\| x\|_ X\). By \(\Lambda_ N\) one denotes an arbitrary subspace of X, such that dim \(\Lambda\) \({}_ N\leq N\). W and V will stand for subsets of X. The N- diameter of Kolmogorov type of W, with respect to V, is given by \(d_ N(W,V,X)=\inf_{\Lambda_ N}\sup_{x\in W}\inf_{y\in \Lambda_ N\cap V}\| x-y\|_ X\). The main result of this paper can be expressed as follows: \(c_ 1N^{-1}\leq d_ N(W^ 1_{\infty},W^ 1_{\infty},L_{\infty})\leq C_ 1N^{-1}\), for some positive constants \(c_ 1\) and \(C_ 1\), and \(c_ 2N^{-2}\leq d_ N(W^ r_{\infty},W^ r_{\infty},L_{\infty})\leq C_ 2N^{-2}\), for \(r=2,3,...\), and certain positive constants \(c_ 2\), \(C_ 2\) that depend on r, but not on N.
Reviewer: C.Corduneanu

MSC:

41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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