Skip to main content
SpringerLink
Log in

Optimal Birkhoff Interpolation and Birkhoff Numbers in Some Function Spaces

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

This paper investigates the optimal Birkhoff interpolation and Birkhoff numbers of some function spaces in space L∞[−1, 1] and weighted spaces Lp,ω[−1,1], 1 ≤ p < ∞, with ω being a continuous integrable weight function in (−1,1). We proved that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal. We also show that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal when the function values of the two endpoints are included in the interpolation systems.

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. Wang H P, Xu G Q. Sampling numbers of a class of infinitely differentiable functions. J Math Anal App, 2020, 484: 123689

    Article  MathSciNet  Google Scholar 

  2. Xu G Q, Wang H. Sample numbers and optimal Lagrange interpolation of Sobolev spaces. Rocky MT J Math, 2021, 51(1): 347–361

    Article  MathSciNet  Google Scholar 

  3. Ben A, Yi S. Compressive Hermite Interpolation: Sparse, High-Dimensional Approximation from Gradient-Augmented Measurements. Constr Approx, 2019, 50: 167–207

    Article  MathSciNet  Google Scholar 

  4. Dell’Accioa F, Tommaso F Di. Complete Hermite-Birkhoff interpolation on scattered data by combined Shepard operators. J Comput Appl Math, 2016, 300: 192–206

    Article  MathSciNet  Google Scholar 

  5. Dell’Accioa F, Tommaso F Di, Nouisser O, Zerroudi B. Fast and accurate scattered Hermite interpolation by triangular Shepard operators. J Comput Appl Math, 2021, 382: 113092

    Article  MathSciNet  Google Scholar 

  6. Garcáa-Marco I, Koiran P. Lower bounds by Birkhoff interpolation. J Complexity, 2017, 39: 38–50

    Article  MathSciNet  Google Scholar 

  7. Goldman G. A case of multivariate Birkhoff interpolation using high order derivatives. J Approx Theory, 2017, 223: 19–28

    Article  MathSciNet  Google Scholar 

  8. Jiao Y J, Wang L L, Huang C. Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis. J Comput Phys, 2016, 305: 1–28

    Article  MathSciNet  Google Scholar 

  9. Mahmoodi A, Nazarzadeh A. A class of Birkhoff type interpolation and applications. Results Math, 2018, 73: 43

    Article  MathSciNet  Google Scholar 

  10. Zare F, Heydari M, Loghmani G B, Wazwaz A-M. Numerical investigation of the Beam-type nano-electrostatic actuator model by using the Birkhoff interpolation method. Int J Appl Comput Math, 2017, 3(Suppl 1): S129–S146

    Article  MathSciNet  Google Scholar 

  11. Allasia G, Cavoretto R, De Rossi A. Hermite-Birkhoff interpolation on scattered data on the sphere and other manifolds. Appl Math Comput, 2018, 318: 35–50

    MathSciNet  MATH  Google Scholar 

  12. Barthelmann V, Novak E, Ritter K. High dimensional polynomial interpolation on sparse grids. Adv Comput Math, 2000, 12: 273–288

    Article  MathSciNet  Google Scholar 

  13. Errachid M, Essanhaji A, Messaoudi A. RMVPIA: a new algorithm for computing the Lagrange multivariate polynomial interpolation. Numer Math, 2020, 84(4): 1507–1534

    MathSciNet  MATH  Google Scholar 

  14. Hinrichs A, Novak E, Ullrich M. On weak tractability of the Clenshaw-Curtis Smolyak algorithm. J Approx Theory, 2014, 183: 31–44

    Article  MathSciNet  Google Scholar 

  15. Irigoyen A. Multidimensional intertwining Leja sequences and applications in bidimensional Lagrange interpolation. J Approx Theory, 2021, 264: 105540

    Article  MathSciNet  Google Scholar 

  16. Xu G Q. On weak tractability of the Smolyak algorithm for approximation problems. J Approx Theory, 2015, 192: 347–361

    Article  MathSciNet  Google Scholar 

  17. Wilson L, Vaughn N, Krasny R. A GPU-accelerated fast multipole method based on barycentric Lagrange interpolation and dual tree traversal. Comput Phys Commun, 2021, 265: 108017

    Article  MathSciNet  Google Scholar 

  18. Liu J, Zhu L Y. Bivariate Lagrange interpolation based on Chebyshev points of the second kind. Acta Math Hung, 2019, 159(2): 618–637

    Article  MathSciNet  Google Scholar 

  19. Hoang N S. On node distributions for interpolation and spectral methods. Math Comp, 2016, 85: 667–692

    Article  MathSciNet  Google Scholar 

  20. Babaev S S, Hayotov A R. Optimal interpolation formulas in \(W_2^{\left({m,m - 1} \right)}\) space. Calcolo, 2019, 56: 23–45

    Article  Google Scholar 

  21. Xu G Q, Liu Z H, Wang H. Sample numbers and optimal Lagrange interpolation of Sobolev spaces \(W_1^r\). Chinese Ann Math, Ser B, 2021, 42(4): 519–528

    Article  MathSciNet  Google Scholar 

  22. Lorentz G G, Jetter K, Riemenschneider S D. Birkhoff interpolation//Encyclopedia of Mathematics and its Applications, Vol 19. Cambridge University Press, 1984

  23. Liu Z H, Lu W T, Xu G Q. Simultaneous approximation of Birkhoff interpolation and the associated sharp inequalities. Int J Wavelets Multi, 2020, 18(4): 2050021

    Article  MathSciNet  Google Scholar 

  24. Nürnberger G. Approximation by Spline Functions. Beijing: Springer-Verlag, 1992

    MATH  Google Scholar 

  25. Novak E, Woźniakowski H. Tractability of Multivariate Problems. Volume I: Linear Information}. Zürich: Eur Math Soc, 2008

    Book  Google Scholar 

  26. DeVore R A, Lorentz G G. Constructive Approximation. New York: Springer-Verlag, 1993

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tianjin Normal University, Tianjin, 300387, China

    Guiqiao Xu

  2. Department of Mathematics, Beijing Normal University, Beijing, 100875, China

    Yongping Liu & Dandan Guo

Authors
  1. Guiqiao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongping Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dandan Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guiqiao Xu.

Additional information

The first author was supported by National Natural Science Foundation of China (11871006, 11671271).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, G., Liu, Y. & Guo, D. Optimal Birkhoff Interpolation and Birkhoff Numbers in Some Function Spaces. Acta Math Sci 43, 125–142 (2023). https://doi.org/10.1007/s10473-023-0108-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-023-0108-5

Key words

2010 MR Subject Classification

Search

Navigation