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On the equality of Kolmogorov and relative widths of classes of differentiable functions. (English) Zbl 1192.46023
This work continues a study by the authors [Math. Notes 65 (5–6), 731–738(1999); translation from Mat. Zametki 65, No. 6, 871–879 (1999; Zbl 0967.42001), and Proceedings of the Steklov Institute of Mathematics 248, 243–254 (2005); translation from Tr. Mat. Inst. Steklova 248, 250–261 (2005; Zbl 1121.41027)] on the widths of a class of differentiable functions whose derivative of order r-1 satisfies a Lipschitz condition with constant M. For some values of M the Kolmogorov width is equal the relative width. The authors ask, “What is the minimal value of M that renders the two widths equal?” Improving a previous result of theirs, the authors sharpen the estimate for such a minimal M.

MSC:
46E15Banach spaces of continuous, differentiable or analytic functions
41A46Approximation by arbitrary nonlinear expressions; widths and entropy