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Ukrainian Mathematical Journal

, Volume 66, Issue 7, pp 994–1012 | Cite as

Interpolation by Splines of Even Degree According to Subbotin and Marsden

  • Yu. S. Volkov
Article

We consider the problem of interpolation by splines of even degree according to Subbotin and Marsden. The study is based on the representation of spline derivatives in the bases of normalized and nonnormalized B-splines. The systems of equations for the coefficients of these representations are obtained. The estimations of the derivatives of the error function in the approximation of an interpolated function by the complete spline are deduced via the norms of inverse matrices of the investigated systems of equations. The relationship between the splines in a sense of Subbotin and the splines in a sense of Marsden is established.

Keywords

Interpolation Spline Interpolation Problem Interpolation Point Divided Difference Interpolate Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    I. J. Schoenberg and A. Whitney, “On P´olya frequency functions, III: The positivity of translation determinants with application to the interpolation problem by spline curves,” Trans. Amer. Math. Soc., 74, No. 2, 246–259 (1953).MathSciNetzbMATHGoogle Scholar
  2. 2.
    J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York (1967).zbMATHGoogle Scholar
  3. 3.
    A. Sharma and A. Meir, “Degree of approximation of spline interpolation,” J. Math. Mech., 15, No. 5, 759–767 (1966).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Yu. N. Subbotin, “On piecewise polynomial interpolation,” Mat. Zametki, 1, No. 1, 63–70 (1967).MathSciNetGoogle Scholar
  5. 5.
    S. B. Stechkin and Yu. N. Subbotin, “Additions,” in: J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications [Russian translation], Mir, Moscow (1972), pp. 270–309.Google Scholar
  6. 6.
    S. B. Stechkin and Yu. N. Subbotin, Splines in Computational Mathematics [in Russian], Nauka, Moscow (1976).Google Scholar
  7. 7.
    M. Marsden, “Quadratic spline interpolation,” Bull. Amer. Math. Soc., 80, No. 5, 903–906 (1974).CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Yu. S. Volkov, “On the construction of interpolation polynomial splines,” Vychislit. Sist., Issue 159, 3–18 (1997).Google Scholar
  9. 9.
    Yu. S. Volkov, “Completely nonnegative matrices in the methods of construction of interpolation splines of odd degree,” Mat. Trudy, 7, No. 2, 3–34 (2004).Google Scholar
  10. 10.
    F. R. Gantmakher and M. G. Krein, Oscillating Matrices and Kernels and Small Oscillations of Mechanical Systems [in Russian], Gostekhizdat, Moscow (1950).Google Scholar
  11. 11.
    C. de Boor and A. Pinkus, “Backward error analysis for totally positive linear systems,” Numer. Math., 27, No. 4, 485–490 (1977).CrossRefzbMATHGoogle Scholar
  12. 12.
    Yu. S. Zav’yalov, B. I. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions [in Russian], Nauka, Moscow (1980).Google Scholar
  13. 13.
    C. de Boor, A Practical Guide to Splines, Springer, New York (1978).CrossRefzbMATHGoogle Scholar
  14. 14.
    Yu. S. Volkov, “On the determination of a complete interpolation spline via B-splines,” Sib. E´lektron. Mat. Izv., 5, 334-338 (2008).zbMATHGoogle Scholar
  15. 15.
    Yu. S. Volkov, “Obtaining a banded system of equations in complete spline interpolation problem via B-spline basis,” Cent. Eur. J. Math., 10, No. 1, 352–356 (2012).CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Yu. S. Volkov, “Investigation of convergence of the process of interpolation for the derivatives of a complete spline,” Ukr. Mat. Visn., 9, No. 2, 278–296 (2012).Google Scholar
  17. 17.
    Yu. S. Volkov and V. T. Shevaldin, “Conditions for the preservation of form in the interpolation by Subbotin and Marsden splines of the second degree,” in: Proc. of the Institute of Mathematics and Mechanics, Ural Division of the Russian Academy of Sciences [in Russian], 18, No. 4, 145–152 (2012).Google Scholar
  18. 18.
    C. de Boor, T. Lyche, and L. L. Schumaker, “On calculating with B-splines, II. Integration,” in: Numerical Methods of Approximation Theory, Birkhäuser, Basel (1976), pp. 123–146.Google Scholar
  19. 19.
    S. Demko, “Inverses of band matrices and local convergence of spline projections,” SIAM J. Numer. Anal., 14, No. 4, 616–619 (1977).CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Yu. S. Volkov
    • 1
  1. 1.Sobolev Institute of MathematicsSiberian Division of the Russian Academy of SciencesNovosibirskRussia

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