Skip to main content
SpringerLink
Log in

Extremal properties of monosplines and best quadrature formulas

  • Published:
Mathematical notes of the Academy of Sciences of the USSR Aims and scope Submit manuscript

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

Literature cited

  1. S. M. Nikol'skii, “Problem of estimates of approximations by quadrature formulas,” Usp. Mat. Nauk,5, No. 2, 165–177 (1950).

    Google Scholar 

  2. G. Peano, “Residuo in formulas de quadratura,” Mathesis,4, No. 34, 5–10 (1914).

    Google Scholar 

  3. S. M. Nikol'skii, Quadrature Formulas [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  4. I. J. Schoenberg, “Spline functions, convex curves and mechanical quadrature,” Bull. Am. Math. Soc.,64, 352–357 (1958).

    Google Scholar 

  5. V. P. Motornyi, “On the best quadrature formula of the form ∑ n i=1 P i f(x i)for certain classes of differentiable periodic functions.Izv. Akad. Nauk SSSR, Ser. Mat.,38, 583–614 (1974).

    Google Scholar 

  6. A. A. Ligun, “Sharp inequalities for spline functions and the best quadrature formulas for certain classes of functions,” Mat. Zametki,16, No. 6, 913–926 (1976).

    Google Scholar 

  7. N. E. Lushpai, “The best quadrature formulas on classes of differentiable periodic functions,” Mat. Zametki,6, No. 4, 475–480 (1969).

    Google Scholar 

  8. N. E. Lushpai, “On an optimal quadrature for a class of differentiable functions,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 4, 57–62 (1973).

    Google Scholar 

  9. N. P. Korneichuk and N. E. Lushpai, “The best quadrature formulas for classes of differentiable functions and piecewise polynomial approximation,” Izv. Akad. Nauk SSSR, Ser. Mat.,33, No. 6, 1416–1437 (1969).

    Google Scholar 

  10. T. N. Busarova, “The best quadrature formulas for a class of differentiable periodic functions,” Ukr. Mat. Zh.,25, No. 3, 291–301 (1973).

    Google Scholar 

  11. Yu. Ya. Doronin, “On the question of the formulas of mechanical quadratures,” in: Collection of the Scientific Works of the Dnepropetrovsk Civil Engineering Institute [in Russian], Nos. 1–2 (1955), pp. 210–217.

    Google Scholar 

  12. T. A. Shaidaeva, “Quadrature formulas with least bound of the remainder for certain classes of functions,” Tr. Mat. Inst. Akad. Nauk SSSR,53, 313–341 (1969).

    Google Scholar 

  13. S. Karlin, “The fundamental theorem of algebra for monosplines satisfying certain boundary conditions and applications to optimal quadrature formulas,” in: Approximation with Special Emphasis on Spline Functions, Academic Press, New York (1969), pp. 467–484.

    Google Scholar 

  14. R. B. Barrar and H. L. Loeb, “On nonlinear characterization problem for monosplines,” J. Approx. Theory,18, 220–240 (1976).

    Google Scholar 

  15. B. D. Boyanov, “Existence of optimal quadrature formulas with preassigned multiplicities of knots,” Mat. Sb.,105, No. 3, 342–370 (1978).

    Google Scholar 

  16. K. Jetter and G. Lange, “Die Eindeutigkeit L2-optimalar polinomialer Monosplines,” Math. Z.,158, 23–34 (1978).

    Google Scholar 

  17. R. S. Johnson, “On monosplines of least deviation,” Trans. Am. Math. Soc.,96, 458–477 (1960).

    Google Scholar 

  18. J. Girschovich, “Extremal properties of Euler-Maclaurin and Grigory quadrature formulas,” Izv. Akad. Nauk Est. SSR, Fiz., Mat.,27, No. 3, 259–265 (1978).

    Google Scholar 

  19. M. Levin, V. Arro, and Yu. Girshovich, “The best quadrature formulas for certain sets of functions,” Izv. Akad. Nauk Est. SSR, Fiz., Mat.,25, No. 2, 112–117 (1976).

    Google Scholar 

  20. S. Karlin and C. Micchelli, “The fundamental theorm of algebra for monosplines satisfying boundary conditions,” Israel J. Math.,11, 405–451 (1972).

    Google Scholar 

  21. A. A. Zhensykbaev, “On the best quadrature formula on the class WrL2,” in: Theory of Approximation of Functions [in Russian], International Conference, Kaluga, Abstracts, Izd. Inst. Mat. Mekh. Ural. Nauk. Tsen. Akad. Nauk SSSR, Sverdlovsk (1975), pp. 48–50.

    Google Scholar 

  22. A. A. Zhensykbaev, “On the best quadrature formula on the class WrLp,” Dokl. Akad. Nauk SSSR,227, No. 2, 277–279 (1976).

    Google Scholar 

  23. A. A. Zhensykbaev [Zensykbaev], “Best quadrature formula for the class WrL2,” Anal. Math.,3, No.1, 83–93 (1977).

    Google Scholar 

  24. A. A. Zhensykbaev, “On the best quadrature formulas for certain classes of nonperiodic functions,” Dokl. Akad. Nauk SSSR,236, No. 3, 531–534 (1977).

    Google Scholar 

  25. A. A. Zhensykbaev, “The best quadrature formula for certain classes of differentiable periodic functions,” Izv. Akad. Nauk SSSR, Ser. Mat.,41, No. 5, 1110–1124 (1977).

    Google Scholar 

  26. A. A. Zhensykbaev, “On the best cubature formulas for certain classes of functions,” in: Mathematics and Mechanics, Abstracts of the Reports at the Sixth Kazakh International Scientific Conference on Mathematics and Mechanics [in Russian], Part l, Matem., Kitap, Alma-Ata (1977), p. 109.

    Google Scholar 

  27. A. A. Zhensykbaev, “On a property of the best quadrature formulas,” Mat. Zametki,23, No. 1, 551–562 (1978).

    Google Scholar 

  28. A. A. Zhensykbaev, “On the best quadrature formulas for certain classes of differentiable functions,” in: Application of the Methods of Function Theory and Functional Analysis to Problems of Mathematical Physics [in Russian], Materials of the Fifth Soviet—Czechoslovakian Conference, Part 2, Izd. Inst. Mat. Sib. Otd. Akad. Nauk SSSR, Novosibirsk (1978), pp. 48–50.

    Google Scholar 

  29. A. A. Zhensykbaev, “Characteristic properties of the best quadrature formulas,” Sib. Mat. Zh.,20, No. 1, 49–68 (1979).

    Google Scholar 

  30. A. A. Zhensykbaev, “Monosplines and the best quadrature formulas for certain classes of nonperiodic functions,” Anal. Math.,5, No. 4, 301–331 (1979).

    Google Scholar 

  31. A. A. Zhensykbaev, “Monosplines with least deviation from zero and the best quadrature formulas,” Dokl. Akad. Nauk SSSR,249, No. 2, 278–281 (1979).

    Google Scholar 

  32. A. A. Zhensykbaev, “Spline functions and the best quadrature formulas,” in: Constructive Theory of Functions [in Russian], Proceeding of the International Conference on Constructive Theory of Functions, 30.5-4.6, Blagoevgrad, 1977, Izd. Akad. Nauk Bolgarii, Sofia (1980), pp. 65–72.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kazakh State University, USSR

    A. A. Zhensykbaev

Authors
  1. A. A. Zhensykbaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Matematicheskie Zametki, Vol. 31, No. 2, pp. 281–298, February, 1982.

The author expresses deep gratitude to his teacher N. P. Korneichuk for discussion of, and assistance with, the obtained results.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhensykbaev, A.A. Extremal properties of monosplines and best quadrature formulas. Mathematical Notes of the Academy of Sciences of the USSR 31, 145–154 (1982). https://doi.org/10.1007/BF01158137

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01158137

Keywords

Search

Navigation