Advertisement

Local exponential splines with arbitrary knots

  • E. V. Strelkova
  • V. T. ShevaldinEmail author
Article

Abstract

We construct local L-splines that have an arbitrary arrangement of knots and preserve the kernel of a linear differential operator L of order r with constant coefficients and real pairwise distinct roots of the characteristic polynomial.

Keywords

local L-splines differential operator arbitrary knots 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Yu. S. Zav’yalov, B. I. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions (Nauka, Moscow, 1980).zbMATHGoogle Scholar
  2. 2.
    T. Lyche and L. L. Schumaker, “Local spline approximation methods,” J. Appr. Theory 15(4), 294–325 (1975).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    E. V. Shevaldina, “Approximation by local L-splines of even order preserving the kernel of a differential operator,” Izv. Tul’sk. Gos. Univ. 2, 62–73 (2009).Google Scholar
  4. 4.
    E. V. Shevaldina, “Local L-splines preserving the kernel of a differential operator,” Sib. Zh. Vychisl. Mat. 13(1), 111–121 (2010).zbMATHGoogle Scholar
  5. 5.
    E. V. Strelkova and V. T. Shevaldin, “Approximation by local L-splines that are exact on subspaces of the kernel of a differential operator,” Proc. Steklov Inst. Math. 273(Suppl. 1), S133–S141 (2011).CrossRefGoogle Scholar
  6. 6.
    E. V. Strelkova and V. T. Shevaldin, “Form preservation under approximation by local exponential splines of an arbitrary order,” Proc. Steklov Inst. Math. 277(Suppl. 1), S171–S180 (2012).CrossRefzbMATHGoogle Scholar
  7. 7.
    Yu. S. Volkov, E. G. Pytkeev, and V. T. Shevaldin, “Orders of approximation by local exponential splines,” Proc. Steklov Inst. Math. 284(Suppl. 1), S175–S184 (2014).CrossRefzbMATHGoogle Scholar
  8. 8.
    V. T. Shevaldin, “Approximation by local parabolic splines with an arbitrary arrangement of knots,” Sib. Zh. Vychisl. Mat. 8(1), 77–88 (2005).zbMATHGoogle Scholar
  9. 9.
    Yu. N. Subbotin, “Approximations by polynomial and trigonometric splines of third order preserving some properties of approximated functions,” Proc. Steklov Inst. Math., Suppl. 2, S231–S242 (2007).Google Scholar
  10. 10.
    Yu. S. Volkov, E. V. Strelkova, and V. T. Shevaldin, “Local approximation by splines with displacement of nodes,” Siberian Adv. Math. 23(1), 69–75 (2011).Google Scholar
  11. 11.
    A. Sharma and I. Cimbalario, “Certain linear differential operators and generalized differences,” Math. Notes 21(2), 91–97 (1977).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    T. Popoviciu, “Sur le reste dans certains formules lineares d’approximation de l’analyse,” Mathematica (Cluj) 1, 95–142 (1959).MathSciNetGoogle Scholar
  13. 13.
    Z. Wronicz, Chebyshevian Splines (Inst. Mat. Polska Akad. Nauk, Warsaw, 1990), Ser. Dissertationes Math., Vol. 305.Google Scholar
  14. 14.
    G. Muhlbach, “A recurrence formula for generalized divided differences and some applications,” J. Approx. Theory 9(2), 165–172 (1973).CrossRefMathSciNetGoogle Scholar
  15. 15.
    G. Walz, “Generalized divided differences, with applications to generalized B-splines,” Calcolo 29(1-2), 111–123 (1992).CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    G. Pólya and G. Szegö,, Problems and Theorems in Analysis (Springer-Verlag, Berlin, 1972; Nauka, Moscow, 1978), Vol. 2.CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Graduate School of Economics and ManagementUral Federal UniversityYekaterinburgRussia

Personalised recommendations