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\(n\)-widths of certain function classes defined by the modulus of continuity. (English) Zbl 1362.41007
The results obtained in this paper are related to the concept of modulus of continuity of fractional order. This concept was introduced almost simultaneously by P. L. Butzer et al. [Can. J. Math. 29, 781–793 (1977; Zbl 0357.26003)] and R. Taberski [Commentat. Math. 19, 389–400 (1977; Zbl 0352.42001)]. This article summarizes and generalizes some of the authors’ results from [Math. Notes 90, No. 5, 748–757 (2011); translation from Mat. Zametki 90, No. 5, 764–775 (2011; Zbl 1284.42003)] and [J. Approx. Theory 164, No. 7, 869–878 (2012; Zbl 1266.42004)] to the case of the modulus of continuity of arbitrary order for classes of differentiable functions (in the sense of Weyl) in the space \(L_2[0,2\pi]\). The authors calculate the exact values of the different \(n\)-widths of these classes with respect to the space \(L_2[0,2\pi]\). In particular, the problem of minimizing the constants in inequalities of Jackson-Stechkin type over all subspaces of dimension \(N\) is solved.
MSC:
41A50 Best approximation, Chebyshev systems
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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