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


1.   Tikkomirov V.M. Approximation theory. In: Analysis II. Encyclopaedia of Mathematical Sciences, vol. 14. Berlin: Springer, 1990, pp. 93–243. doi: 10.1007/978-3-642-61267-1_2 

2.   Konovalov V.N. Estimates of Kolmogorov-type widths for classes of differentiable periodic functions. Math. Notes, 1984, vol. 35, no. 3, pp. 193–199. doi: 10.1007/BF01139916 

3.   Subbotin Yu.N., Telyakovskii S.A. Exact values of relative widths of classes of differentiable functions. Math. Notes, 1999, vol. 65, no. 6, pp. 731–738. doi: 10.1007/BF02675588 

4.   Tikhomirov V.M. Some remarks on relative diameters. In: “Approximation and function spaces”, Proc. 27th Semest., Warsaw/Pol. 1986. Banach Center Publications, vol. 22. Warszawa: PWN, 1989, pp. 471–474.

5.   Babenko V.F. On best uniform approximations by splines in the presence of restrictions on their derivatives. Math. Notes, 1991, vol. 50, no. 6, pp. 1227–1232. doi: 10.1007/BF01158262 

6.   Malykhin Yu.V. Relative widths of Sobolev classes in the uniform and integral metrics. Proc. Steklov Inst. Math., 2016, vol. 293, pp. 209–215. doi: 10.1134/S0081543816040155 

7.   DeVore R.A., Lorentz G.G. Constructive approximation. Berlin; Heidelberg: Springer, 1993, 452 p. doi: 10.1007/978-3-662-02888-9 

8.   Malykhin Yuri. Widths and rigidty [e-resource]. 2022. 33 p. Available on: https://arxiv.org/pdf/2205.03453.pdf 

9.   Ismagilov R.S. On n-dimensional diameters of compacts in a Hilbert space. Funct. Anal. Appl., 1968, vol. 2, no. 2, pp. 125–132. doi: 10.1007/BF01075946 

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.