Skip to main content

Asymptotic estimates for the approximation and entropy numbers of a one-weight Riemann-Liouville operator

Abstract

We obtain two-sided estimates for the asymptotic behavior of the approximation and entropy numbers of a one-weight Riemann-Liouville operator of an arbitrary integer order acting in Lebesgue spaces on the semiaxis.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    B. Carl, “Entropy numbers of diagonal operators with an application to eigenvalue problems,” J. Approx. Theory 32(2), 135–150 (1981).

    MATH  Article  MathSciNet  Google Scholar 

  2. 2.

    B. Carl and I. Stephany, Entropy, Compactness and the Approximation of Operators, vol. 98 of Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 1990).

    Google Scholar 

  3. 3.

    D. E. Edmunds, W. D. Evans, and D. J. Harris, “Approximation numbers of certain Volterra integral operators,” J. London Math. Soc. (2) 37(3), 471–489 (1988).

    MATH  Article  MathSciNet  Google Scholar 

  4. 4.

    D. E. Edmunds, W. D. Evans, and D. J. Harris, “Two-sided estimates of the approximation numbers of certain Volterra integral operators,” Studia Math. 124(1), 59–80 (1997).

    MATH  MathSciNet  Google Scholar 

  5. 5.

    D. E. Edmunds and H. Tribel, Function Spaces, Entropy Numbers, Differential Operators, vol. 120 of Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 1996).

    Google Scholar 

  6. 6.

    E. D. Gluskin, “Norms of random matrices and diameters of finite-dimensional sets,” Mat. Sb. (N.S.) 120(2), 180–189 (1983).

    MathSciNet  Google Scholar 

  7. 7.

    M. A. Lifshits and W. Linde, “Approximation and entropy numbers of Volterra operators with application to Brownian motion,” Mem. Amer. Math. Soc. 157(745), viii+87 (2002).

    MathSciNet  Google Scholar 

  8. 8.

    R. Linde, in Proceedings of the 13th Winter School on Abstract Analysis (Srní, 1985) (1986), 10, pp. 83–110.

    Google Scholar 

  9. 9.

    E. N. Lomakina and V. D. Stepanov, “On asymptotic behavior of the approximation numbers and estimates of Schatten-von Neumann norms of the Hardy-type integral operators,” in Function Spaces and Applications (Delhi, 1997) (Narosa, New Delhi, 2000), pp. 153–187.

    Google Scholar 

  10. 10.

    J. Newman and M. Solomyak, “Two-sided estimates on singular values for a class of integral operators on the semi-axis,” Integral Equations Operator Theory 20(3), 335–349 (1994).

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    A. Pietsch, Operator Ideals (Mir, Moscow, 1982) [in Russian].

    MATH  Google Scholar 

  12. 12.

    D. V. Prokhorov, “On the boundedness and compactness of a class of integral operators,” J. London Math. Soc. (2) 61(2), 617–628 (2000).

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    D. V. Prokhorov and V. D. Stepanov, “Weighted estimates for Riemann-Liouville operators and their applications,” Trudy Mat. Inst. Steklov 243(4), 289–312 (2003) [Proc. Steklov Inst. Math. 243 (4), 278–301 (2003)].

    MathSciNet  Google Scholar 

  14. 14.

    M. Solomyak, “Estimates for the approximation numbers of the weighted Riemann-Liouville operator in the spaces L p,” in Complex Analysis, Operators, and Related Topics (Birkhäuser, Basel, 2000), vol. 113 of Oper. Theory Adv. Appl., pp. 371–383.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Dedicated to Yu. G. Reshetnyak on the occasion of his 75th birthday

Original Russian Text © E. N. Lomakina and V. D. Stepanov, 2006, published in Matematicheskie Trudy, 2006, Vol. 9, No. 1, pp. 52–100.

About this article

Cite this article

Lomakina, E.N., Stepanov, V.D. Asymptotic estimates for the approximation and entropy numbers of a one-weight Riemann-Liouville operator. Sib. Adv. Math. 17, 1–36 (2007). https://doi.org/10.3103/S1055134407010014

Download citation

Key words

  • integral Riemann-Liouville operator
  • approximation and entropy numbers
  • Lebesgue space