Skip to main content
SpringerLink
Log in

Biorthogonal systems of functionals and decomposition matrices for minimal splines

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

For the minimal splines of arbitrary order on a nonuniform grid, a system of linear functionals biorthogonal to the system of coordinate splines is constructed. The matrices of refining and sparsing decompositions are obtained for the spaces of splines of arbitrary order associated with infinite and finite nonuniform grids on an interval and on a segment, respectively.

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

Similar content being viewed by others

References

  1. Yu. K. Dem’yanovich, “On the construction of the spaces of local functions on a nonuniform grid,” Zap. Nauch. Sem. LOMI AN SSSR, 124, 140–163 (1983).

    MathSciNet  MATH  Google Scholar 

  2. Yu. K. Dem’yanovich, Local Approximation on a Manifold and Minimum Splines [in Russian], St.-Petersburg Univ., St.-Petersburg, 1994.

    Google Scholar 

  3. Yu. K. Dem’yanovich and O. M. Kosogorov, “On the calculation of decomposition matrices in the splinewavelet expansion,” in Methods of Calculations [in Russian], St.-Petersburg Univ., St.-Petersburg, 2009, pp. 71–97.

    Google Scholar 

  4. Yu. K. Dem’yanovich and A. A. Makarov, “Gauge relations for nonpolynomial splines,” J. Math. Sci., 142, No. 1, 1769–1787 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. A. Makarov, “The reconstruction matrices and gauge relations for minimum splines,” J. Math. Sci., 178, No. 6, 605—621 (2011).

    Article  Google Scholar 

  6. A. A. Makarov, “Piecewise continuous spline-wavelets on a nonuniform grid,” Trudy SPIIRAN, Issue 14, 103—131 (2010).

    Google Scholar 

  7. A. A. Makarov, “On the construction of splines with maximum smoothness,” J. Math. Sci., 178, No. 6, 589—604 (2011).

    Article  Google Scholar 

  8. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, New York, 2008.

    Google Scholar 

  9. V. S. Ryaben’kii, On the Stability of Finite-Difference Equations [in Russian], Candidate degree thesis (Phys.-Math. Sci.), Moscow, 1952.

  10. M. Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, Addison-Wesley, Boston, MA, 1995.

    Google Scholar 

  11. V. P. Zastavnyi and R. M. Trigub, “Positive definite splines of a special form,” Mat. Sb., 193:12, 41–68 (2002).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St.-Petersburg State University, Faculty of Mathematics and Mechanics, 28, Universitetskii pr., St.-Petersburg, 198504, Petrodvorets, Russia

    Anton A. Makarov

Authors
  1. Anton A. Makarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anton A. Makarov.

Additional information

To Roal’d Mikhailovich Trigub by his 75-th anniversary

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 9, No. 2, pp. 219–236, April–May, 2012.

The work is partially supported by a grant of RFFS (10-01-00245) and by a grant of the President of the Russian Federation (MK-5219.2011.1).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Makarov, A.A. Biorthogonal systems of functionals and decomposition matrices for minimal splines. J Math Sci 187, 57–69 (2012). https://doi.org/10.1007/s10958-012-1049-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-012-1049-z

Keywords

Search

Navigation