Skip to main content
SpringerLink
Log in

The General Problem of Polynomial Spline Interpolation

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We study the general problem of interpolation by polynomial splines and consider the construction of such splines using the coefficients of expansion of a certain derivative in B-splines. We analyze the properties of the obtained systems of equations and estimate the interpolation error.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967; Mir, Moscow, 1972).

    MATH  Google Scholar 

  2. Yu. S. Volkov, “Interpolation by splines of even degree according to Subbotin and Marsden,” Ukr. Math. J. 66 (7), 994–1012 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  3. S. B. Stechkin and Yu. N. Subbotin, Splines in Numerical Mathematics (Nauka, Moscow, 1976) [in Russian].

    MATH  Google Scholar 

  4. Yu. S. Zav’yalov, B. I. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  5. Yu. S. Volkov, “Divergence of interpolating splines of odd degree,” in Computational Systems (IM SO AN SSSR, Novosibirsk, 1984), Issue 106, pp. 41–56 [in Russian].

    Google Scholar 

  6. Yu. S. Volkov, “On the construction of interpolating polynomial splines,” in Computational Systems, Issue 159: Spline Functions and Their Applications (IM SO RAN, Novosibirsk, 1997), pp. 3–18 [in Russian].

    Google Scholar 

  7. Yu. S. Volkov, “A new method for constructing cubic interpolating splines,” Dokl. Math. 65 (1), 13–15 (2002).

    MATH  Google Scholar 

  8. Yu. S. Volkov, “Totally positive matrices in the methods of constructing interpolation splines of odd degree,” Siberian Adv. Math. 15 (4), 96–125 (2005).

    MathSciNet  MATH  Google Scholar 

  9. Yu. S. Volkov, “Unconditional convergence of one more middle derivative for odd degree spline interpolation,” Dokl. Math. 71 (2), 250–252 (2005).

    MATH  Google Scholar 

  10. Yu. S. Volkov, “Convergence analysis of an interpolation process for the derivatives of a complete spline,” J. Math. Sci. 187 (1), 101–114 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  11. I. J. Schoenberg and A. Whitney, “On Pólya frequency functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves,” Trans. Amer. Math. Soc. 74 (2), 246–259 (1953).

    MathSciNet  MATH  Google Scholar 

  12. C. de Boor, A Practical Guide to Splines (Springer, New York, 1978; Radio i Svyaz’, Moscow, 1985).

    MATH  Google Scholar 

  13. C. de Boor, T. Lyche, and L. L. Schumaker, “On calculating with B-splines. II. Integration,” in Numerische Methoden der Approximationstheorie, Vol. 3, Ed. by L. Collatz, H. Werner, and G. Meinardus (Birkhäuser, Basel, 1976), International Series of Numerical Mathematics 30, pp. 123–146.

    Article  MathSciNet  MATH  Google Scholar 

  14. F. R. Gantmakher and M. G. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (Gostekhizdat, Moscow, 1950; Amer. Math. Soc., Providence, RI, 2002).

    Google Scholar 

  15. C. de Boor and A. Pinkus, “Backward error analysis for totally positive linear systems,” Numer. Math. 27 (4), 485–490 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  16. L. L. Schumaker, Spline Functions: Basic Theory (Wiley, New York, 1981).

    MATH  Google Scholar 

  17. Yu. S. Volkov and V. T. Shevaldin, “Shape preserving conditions for quadratic spline interpolation in the sense of Subbotin and Marsden,” Trudy Inst. Mat. Mekh. UrO RAN 18 (4), 145–152 (2012).

    Google Scholar 

  18. C. de Boor, “On the convergence of odd-degree spline interpolation,” J. Approx. Theory 1 (4), 452–463 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  19. Yu. S. Volkov and V. L. Miroshnichenko, “Norm estimates for the inverses of matrices of monotone type and totally positive matrices,” Sib. Math. J. 50 (6), 982–987 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  20. C. de Boor, “On bounding spline interpolation,” J. Approx. Theory 14 (3), 191–203 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  21. Yu. S. Volkov and Yu. N. Subbotin, “Fifty years of Schoenberg’s problem on the convergence of spline interpolation,” Proc. Steklov Inst. Math. 288 (Suppl. 1), S222–S237 (2015).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090, Russia

    Yu. S. Volkov

  2. Novosibirsk State University, Novosibirsk, 630090, Russia

    Yu. S. Volkov

Authors
  1. Yu. S. Volkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. S. Volkov.

Additional information

Original Russian Text © Yu.S. Volkov, 2016, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, Vol. 22, No. 4, pp. 114–125.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Volkov, Y.S. The General Problem of Polynomial Spline Interpolation. Proc. Steklov Inst. Math. 300 (Suppl 1), 187–198 (2018). https://doi.org/10.1134/S0081543818020190

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543818020190

Keywords

Search

Navigation