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On the equality of Kolmogorov and relative widths of classes of differentiable functions. (English. Russian original) Zbl 1192.46023
Math. Notes 86, No. 3, 432-439 (2009); translation from Mat. Zametki 86, No. 3, 456-465 (2009); erratum Math. Notes 87, No. 3, 452 (2010).
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:
46E15 Banach spaces of continuous, differentiable or analytic functions
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
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[1] Subbotin, Yu. N.; Telyakovskii, S. A., Exact values of relative widths of classes of differentiable functions, Mat. Zametki, 65, 871-879, (1999)
[2] Subbotin, Yu. N.; Telyakovskii, S. A., On relative widths of classes of differentiable functions, 250-261, (2005), Moscow · Zbl 1121.41027
[3] N. P. Korneichuk, Extremal Problems of Approximation Theory (Nauka, Moscow, 1976) [in Russian].
[4] Telyakovskii, S. A., An estimate, useful in problems of approximation theory, of the norm of a function by means of its Fourier coefficients, 65-97, (1971), Moscow
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