Skip to main content
SpringerLink
Log in

A generalisation of de la Vallée-Poussin procedure to multivariate approximations

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

The theory of Chebyshev approximation has been extensively studied. In most cases, the optimality conditions are based on the notion of alternance or alternating sequence (that is, maximal deviation points with alternating deviation signs). There are a number of approximation methods for polynomial and polynomial spline approximation. Some of them are based on the classical de la Vallée-Poussin procedure. In this paper we demonstrate that under certain assumptions the classical de la Vallée-Poussin procedure, developed for univariate polynomial approximation, can be extended to the case of multivariate approximation. There are two main advantages of our approach. First of all, it provides an elegant geometrical interpretation of the procedure. Second, the corresponding basis functions are not restricted to be monomials and therefore can be extended to a larger family of functions.

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. Nürnberger, G.: Approximation by spline functions, Springer-Verlag, 1989

  2. Rice, J.: Characterization of Chebyshev approximation by splines, SIAM Journal on Numerical Analysis 4 (1967), no. 4, 557–567, (1967)

  3. Schumaker, L.: Uniform approximation by Chebyshev spline functions. II: free knots, SIAM Journal of Numerical Analysis 5, 647–656, (1968)

  4. Rice, John: Tchebycheff approximation in several variables. Trans. Amer. Math. Soc. 109, 444–466 (1963)

    Article  MathSciNet  Google Scholar 

  5. Charles Jean de la Vallée Poussin: Sur la méthode de l’approximation minimum. Annales de la Société Scientifique de Bruxelles 35, 1–16 (1911)

    MATH  Google Scholar 

  6. Sukhorukova, Nadezda: Uniform approximation by the highest defect continuous polynomial splines: necessary and sufficient optimality conditions and their generalisations. Journal of Optimization Theory and Applications 147(2), 378–394 (2010)

    Article  MathSciNet  Google Scholar 

  7. Peiris, V., Sharon, N., Sukhorukova, N. and Ugon, J.: Generalised rational approximation and its application to improve deep learning classifiers, Applied Mathematics and Computation 389, 125560 (2021)

  8. Blair, J.M., Edwards, C.A., Johnson, J.H.: Rational Chebyshev Approximations for the Inverse of the Error Function. Mathematics of Computation 30(136), 827–830 (1976)

    Article  MathSciNet  Google Scholar 

  9. Nakatsukasa, Yuji, Sete, Olivier, Trefethen, Lloyd N.: The aaa algorithm for rational approximation. SIAM Journal on Scientific Computing 40(3), A1494–A1522 (2018)

    Article  MathSciNet  Google Scholar 

  10. Zalinescu, C.: Convex analysis in general vector spaces, World Scientific, (2002)

  11. Sukhorukova, N., Ugon, J. and Yost, D.: Chebyshev multivariate polynomial approximation: alternance interpretation, in 2016 MATRIX Annuals , David R. Wood, Jan de Gier, Cheryl E. Praeger and Terence Tao (eds.), Springer International Publishing, Switzerland, : 177–182, (2018)

  12. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer-Verlag, Berlin, Heidelberg, New York (1993)

    Book  Google Scholar 

  13. Bertsekas, D.: Convex Analysis and Optimization Chapter 1 Solutions, Athena Scientific (2008)

  14. Radon, J.: Mengen konvexer körper, die einen gemeinsamen punkt enthalten. Mathematische Annalen 83(1–2), 113–115 (1921)

    Article  MathSciNet  Google Scholar 

  15. Sukhorukova, Nadezda: Vallée Poussin theorem and Remez algorithm in the case of generalised degree polynomial spline approximation. Pacific Journal of Optimization 6(2), 103–114 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was supported by the Australian Research Council (ARC), Solving hard Chebyshev approximation problems through nonsmooth analysis (Discovery Project DP180100602). This paper was inspired by the discussions during a recent MATRIX program “Approximation and Optimisation” that took place in July 2016. We are thankful to the MATRIX organisers, support team and participants for a terrific research atmosphere and productive discussions.

Author information

Authors and Affiliations

  1. Swinburne University of Technology, John St, Hawthorn, VIC 3122, Australia

    Nadezda Sukhorukova

  2. 221 Burwood Highway, Burwood, VIC 3125, Australia

    Julien Ugon

Authors
  1. Nadezda Sukhorukova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julien Ugon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nadezda Sukhorukova.

Additional information

Communicated by Thanh Tran

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sukhorukova, N., Ugon, J. A generalisation of de la Vallée-Poussin procedure to multivariate approximations. Adv Comput Math 48, 5 (2022). https://doi.org/10.1007/s10444-021-09919-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-021-09919-x

Keywords

PACS:

Search

Navigation