Skip to main content
SpringerLink
Log in

Widths of weighted Sobolev classes with constraints f(a) = · · · = f(k-1)(a) = f(k)(b) = · · · = f(r-1)(b) = 0 and the spectra of nonlinear differential equations

  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper we prove that the Kolmogorov widths of the weighted Sobolev class W p,g r,k [a, b] with restrictions f(a) = · · · = f(k-1)(a) = f(k)(b) = · · · = f(r-1)(b) = 0 in the weighted Lebesgue space L q,v [a, b] coincide with the spectral numbers of some nonlinear differential equation. We assume that 1 < q ≤ p < ∞, the weights are positive almost everywhere, and the Sobolev class is compactly embedded in the space L q,v [a, b].

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

Access this article

Log in via an institution

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. V. F. Babenko and N. V. Parfinovich, “Exact Values of Best Approximations for Classes of Periodic Functions by Splines of Deficiency 2,” Math. Notes 85 (3), 515–527 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  2. V. F. Babenko and N. V. Parfinovich, “On the Exact Values of the Best Approximations of Classes of Differentiable Periodic Functions by Splines,” Math. Notes 87 (5), 623–635 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. P. Buslaev, “Extremal Problems in the Theory of Approximations and the Nonlinear Oscillations,” DAN SSSR 305 (6), 1289–1294 (1989) (in Russian).

    Google Scholar 

  4. A. P. Buslaev, “On the Asymptotics of Widths and Spectra of Nonlinear Differential Equations,” Algebra i Analiz 3 (6), 108–118 (1991) [St. Petersburg Math. J. 3 (6), 1303–1312 (1991)].

    MathSciNet  MATH  Google Scholar 

  5. A. P. Buslaev and V. M. Tikhomirov, “Spectra of Nonlinear Differential Equations and Widths of Sobolev Classes,” Mat. Sb. 181 (12), 1587–1606 (1990) [Math. USSR-Sb. 71 (2), 427–446 (1992)].

    MathSciNet  MATH  Google Scholar 

  6. W. D. Evans, D. J. Harris, and J. Lang, “The Approximation Numbers of Hardy-Type Operators on Trees,” Proc. London Math. Soc. (3) 83 (2), 390–418 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  7. D. E. Edmunds, R. Kerman, and J. Lang, “Remainder Estimates for the Approximation Numbers of Weighted Hardy Operators Acting on L 2”, J. Anal. Math. 85 (1), 225–243 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  8. D. E. Edmunds and J. Lang, “Approximation Numbers and Kolmogorov Widths of Hardy-Type Operators in a Non-Homogeneous Case,” Math. Nachr. 297 (7), 727–742 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  9. D. E. Edmunds and J. Lang, “Asymptotics for Eigenvalues of a Non-Linear Integral System,” Boll. Unione Mat. Ital. 1 (1), 105–119 (2008).

    MathSciNet  MATH  Google Scholar 

  10. D. E. Edmunds and J. Lang, “Coincidence of Strict s-Numbers of Weighted Hardy Operators,” J. Math. Anal. Appl. 381 (2), 601–611 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  11. D. E. Edmunds and J. Lang, “Asymptotic Formulae for S-Numbers of a Sobolev Embedding and a Volterra Type Operator,” Rev. Mat. Compl. 29 (1), 1–11 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucia, J. Pelant, and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry (Springer, New York, 2001).

    Book  MATH  Google Scholar 

  13. S. Heinrich, “On the Relation Between Linear N-Widths and Approximation Numbers,” J. Approx. Theory 58 (3), 315–333 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Karlin, and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Pure and Applied Mathematics, Vol. XV (Interscience Publishers John Wiley & Sons, New York–London–Sydney, 1966; Nauka, Fizmatlit, Moscow, 1976).

    MATH  Google Scholar 

  15. A. N. Kolmogorov, “Über die beste Annäherung von Funktion einer gegebenen Funktionenklasse,” Ann. Math. 37, 107–110 (1936).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Kufner and H. P. Heinig, “The Hardy Inequality for Higher-Order Derivatives,” in: Differential Equations and Function Spaces, Collection of Papers Dedicated to the Memory of Academician S. L. Sobolev, Trudy Mat. Inst. Steklov 192, 105–113 (1990) [Proc. Steklov Inst. Math., 192, 113–121 (1992)].

    MathSciNet  MATH  Google Scholar 

  17. J. Lang, “Improved Estimates for the Approximation Numbers of Hardy-Type Operators,” J. Appr. Theory 121 (1), 61–70 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. A. Ligun, “Diameters of Certain Classes of Differentiate Periodic Functions,” Math. Notes 27 (1), 34–41 (1980).

    Article  MATH  Google Scholar 

  19. J. I. Makovoz, “On a Method for Estimation from Below of Diameters of Sets in Banach Spaces,” Mat. Sb. 87(129) (1), 136–142 (1972) [Math. USSR-Sb. 16 (1), 139–146 (1972)].

    MathSciNet  MATH  Google Scholar 

  20. Yu. V. Malykhin, “Asymptotic Properties of Chebyshev Splines with Fixed Number of Knots,” Fund Prikl. Mat. 19 (5), 143–166 (2014) [J. Math. Sci., New York 218 (5), 647–663 (2016)].

    MathSciNet  Google Scholar 

  21. Yu. V. Malykhin, “Relative Widths of Sobolev Classes in the Uniform and Integral Metrics,” Tr. Mat. Inst. Steklova 293, 217–223 (2016) [Proc. Steklov Inst. Math. 293, 209–215 (2016)].

    MathSciNet  MATH  Google Scholar 

  22. A. Pietsch, “s-Numbers of Operators in Banach Space,” Studia Math. 51, 201–223 (1974).

    MathSciNet  MATH  Google Scholar 

  23. A. Pinkus, “On n-Widths of Periodic Functions,” J. Anal. Math. 35, 209–235 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Pinkus, “n-Widths of Sobolev Classes in L p,” Constructive Approximation 1 (1), 15–62 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  25. V. D. Stepanov, “Two-Weighted Estimates for Riemann-Liouville Integrals,” Izv. Akad. Nauk SSSR Ser. Mat. 54 (3), 645–656 (1990) [Math. USSR-Izv. 36 (3), 669-681 (1991)].

    MATH  Google Scholar 

  26. V. D. Stepanov, “Weighted Norm Inequalities of Hardy Type for a Class of Integral Operators,” J. London Math. Soc. 50 (1), 105–120 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  27. V. M. Tikhomirov, “Theory of Approximations,” In: Current problems in mathematics. Fundamental directions 14 (Itogi Nauki i Tekhniki) (Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987), pp. 103–260 [Encycl. Math. Sci. 14, 93–243 (1990)].

    MATH  Google Scholar 

  28. V. M. Tikhomirov, “Diameters of Sets in Functional Spaces and the Theory of Best Approximations,” Russian Math. Surveys 15 (3), 75–111 (1960).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. V. M. Tihomirov [Tikhomirov] and S. B. Babadzanov, “Diameters of a Function Class in an L p-Space (p ≥ 1),” Izv. Akad. Nauk UzSSR Ser. Fiz. Mat. Nauk 11, 24–30 (1967) (Russian).

    MathSciNet  Google Scholar 

  30. V. M. Tikhomirov, “Some Problems in Approximation Theory,” Math. Notes 9 (5), 343–350 (1971).

    Article  Google Scholar 

  31. A. A. Vasil’eva, “Widths of Weighted Sobolev Classes on a Closed Interval and the Spectra of Nonlinear Differential Equations,” Russian J. Math. Phys. 17 (3), 363–393 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  32. A. A. Vasil’eva, “Kolmogorov Widths and Approximation Numbers of Sobolev Classes with Singular Weights,” Algebra i Analiz 24 (1), 3–39 (2012).

    MathSciNet  Google Scholar 

  33. A. A. Vasil’eva, “Estimates for the Widths of Weighted Sobolev Classes,” Sbornik: Math. 201 (7), 947–984 (2010).

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Russia, 119991, Moscow, GSP-1, 1 Leninskiye Gory, Main Building

    A. A. Vasil’eva

Authors
  1. A. A. Vasil’eva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Vasil’eva.

Additional information

The research was financially supported by the Russian Foundation for Basic Research (grant no. 16-01-00295).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vasil’eva, A.A. Widths of weighted Sobolev classes with constraints f(a) = · · · = f(k-1)(a) = f(k)(b) = · · · = f(r-1)(b) = 0 and the spectra of nonlinear differential equations. Russ. J. Math. Phys. 24, 376–398 (2017). https://doi.org/10.1134/S1061920817030116

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920817030116

Search

Navigation