Yu.V. Malykhin. A complete description of the relative widths of Sobolev classes in the uniform metric ... P. 137-142

We consider the width of the Sobolev class of $2\pi$-periodic functions with $\|f^{(r)}\|_\infty\le 1$ with respect to the set of functions $g$ such that $\|g^{(r)}\|_\infty\le M$ in the uniform metric $K_n:=K_n(W^r_\infty,MW^r_\infty,L_\infty)$. We prove a lower bound on $K_n$ for $M=1+\varepsilon$ with small $\varepsilon$. This bound together with earlier results completes the anаlysis of the behaviour of $K_n$.

Keywords: Kolmogorov and relative widths

Received June 7, 2022

Revised August 24, 2022

Accepted August 29, 2022

Funding Agency: This research was carried out at Lomonosov Moscow State University with the financial support of the Russian Science Foundation (project no. 22-11-00129).

Yuriy Vyacheslavovich Malykhin, Cand. Sci. (Phys.-Math.), Steklov Mathematical Institute of the Russian Academy of Sciences; Lomonosov Moscow State University, Moscow, 119991 Russia, e-mail: malykhin@mi-ras.ru

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Cite this article as: Yu.V. Malykhin. A complete description of the relative widths of Sobolev classes in the uniform metric, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2022, vol. 28, no. 4, pp. 137–142; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S188–S192.