Skip to main content
Log in

Approximation of Sobolev Classes by Their Finite-Dimensional Sections

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We consider relative widths characterizing the best approximation of a fixed set by its sections of given dimension. For Sobolev classes of periodic functions of a single variable with constraints inL orL 1 on higher-order derivatives, we present the exact orders of such widths in the spaces L q.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. N. Kolmogoroff, “Ñber die beste Annäherung von Funktionen einer gegebenen Funktionenklasse,” Ann. Math., 37 (1936), no. 1, 107-110.

    Google Scholar 

  2. N. P. Korneichuk, Extremum Problems of Approximation Theory [in Russian], Nauka, Moscow, 1976.

    Google Scholar 

  3. V. M. Tikhomirov, Some Problems of Approximation Theory [in Russian], Moskov. Gos. Univ., Moscow, 1976.

    Google Scholar 

  4. A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  5. N. P. Korneichuk, Sharp Constants in Approximation Theory [in Russian], Nauka, Moscow, 1987.

    Google Scholar 

  6. V. M. Tikhomirov, “Approximation Theory,” in: Current Problems in Mathematics. Fundamental Directions [in Russian], vol. 14, Itogi Nauki i Tekhniki, VINITI, Academy of Sciences of the USSR, Moscow, 1987, pp. 103-260.

    Google Scholar 

  7. V. N. Konovalov, “Estimates of Kolmogorov-type widths for classes of differentiable periodic functions,” Mat. Zametki [Math. Notes], 35 (1984), no. 3, 369-380.

    Google Scholar 

  8. S. B. Stechkin, “On best approximation of given classes of functions by any polynomials,” Uspekhi Mat. Nauk [Russian Math. Surveys], 9 (1954), no. 1, 133-134.

    Google Scholar 

  9. V. M. Tikhomirov, “Widths of sets in function spaces and best approximation theory,” Uspekhi Mat. Nauk [Russian Math. Surveys], 15 (1960), no. 3, 81-120.

    Google Scholar 

  10. V. M. Tikhomirov, “The best methods of approximation and interpolation of differentiable functions in C[?1, 1],” Mat. Sb. [Math. USSR-Sb.], 80 (1969), no. 2, 290-304.

    Google Scholar 

  11. V. M. Tikhomirov, “Some remarks on relative diameters,” Banach Center Publ., 22 (1989), 471-474.

    Google Scholar 

  12. V. F. Babenko, “Approximations in mean in the presence of constraints on the derivatives of approximating functions,” in: Problems of Analysis and Approximation [in Russian], A Collection of Papers, IM AN UkrSSR, Kiev, 1989, pp. 9-18.

    Google Scholar 

  13. V. F. Babenko, “On the relative widths of function classes with bounded mixed derivative,” East J. Approximation, 2 (1996), no. 3, 319-330.

    Google Scholar 

  14. V. F. Babenko, “On best uniform approximations by splines in the presence of constraints on their the derivatives,” Mat. Zametki [Math. Notes], 50 (1991), no. 6, 24-30.

    Google Scholar 

  15. V. F. Babenko, “On best L 1-approximations by splines in the presence of constraints on their derivatives,” Mat. Zametki [Math. Notes], 51 (1992), no. 5, 12-19.

    Google Scholar 

  16. Yu. N. Subbotin, “Inheriting the properties of monotonicity and convexity in local approximation,” Zh. Vychisl. Mat. i Mat. Fiz. [Comput. Math. and Math. Phys.], 33 (1993), 996-1003.

    Google Scholar 

  17. Yu. N. Subbotin and S. A. Telyakovskii, “Exact values of relative widths of classes of differentiable functions,” Mat. Zametki [Math. Notes], 65 (1999), no. 6, 871-879.

    Google Scholar 

  18. Yu. N. Subbotin and S. A. Telyakovskii, “Relative widths of classes of differentiable functions in the metric of L2,” Uspekhi Mat. Nauk [Russian Math. Surveys], 56 (2001), no. 4, 159-160.

    Google Scholar 

  19. L. É. Azar, Approximation of Function Classes in the Presence of Constraints [in Russian], Kandidat thesis in the physico-mathematical sciences, DGU, Dnepropetrovsk, 1999.

    Google Scholar 

  20. R. S. Ismagilov, “On n-dimensional widths of compact sets in Hilbert space,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 2 (1968), no. 2, 32-39.

    Google Scholar 

  21. V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow, 1977.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Konovalov, V.N. Approximation of Sobolev Classes by Their Finite-Dimensional Sections. Mathematical Notes 72, 337–349 (2002). https://doi.org/10.1023/A:1020547320561

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020547320561

Navigation