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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
##### Citations:
Zbl 0967.42001; Zbl 1121.41027; Zbl 1202.46028
Full Text:
##### References:
 [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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