Skip to main content
SpringerLink
Log in

On summability of weighted Lagrange interpolation. II

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

The aim of this paper is to continue our investigations started in [15], where we studied the summability of weighted Lagrange interpolation on the roots of orthogonal polynomials with respect to a weight function w. Starting from the Lagrange interpolation polynomials we constructed a wide class of discrete processes which are uniformly convergent in a suitable Banach space (C ρ, ‖·‖ρ) of continuous functions (ρ denotes (another) weight). In [15] we formulated several conditions with respect to w, ρ, (C ρ, ‖·‖ρ) and to summation methods for which the uniform convergence holds. The goal of this part is to study the special case when w and ρ are Freud-type weights. We shall show that the conditions of results of [15] hold in this case. The order of convergence will also be considered.

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.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. G. Criscuolo, B. Della Vecchia, D. Lubinsky and G. Mastroianni, Functions of second kind for Freud weights, J. Math. Anal. Appl., 189 (1994), 256-296.

    Article  MathSciNet  Google Scholar 

  2. Z. Ditzian and D. S. Lubinsky, Jackson and smoothness theorems for Freud weights, Constr. Approx., 13 (1997), 99-152.

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Freud, Orthogonal Polynomials, Akadémiai Kiadó (Budapest, 1971).

    Google Scholar 

  4. G. Freud, On Markov-Bernstein-type inequalities and their applications, J. Approx. Theory, 19 (1977), 22-37.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. L. Levin and D. S. Lubinsky, Christoffel functions, orthogonal polynomials, and Nevai's conjecture for Freud weights, Constr. Approx., 8 (1992), 463-535.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Levin and D. S. Lubinsky, Orthogonal Polynomials for Exponential Weights, Springer-Verlag (New York-Berlin-Heidelberg, 2001).

    Google Scholar 

  7. D. S. Lubinsky, Jackson and Bernstein theorems for exponential weights, Bolyai Society Mathematical Studies, 5, Approximation Theory and Function Series, Budapest (Hungary), (P. Vértesi et al., eds.), Budapest, 1996, 117-139.

  8. D. S. Lubinsky, An extension of the Erdős-Turán inequality for the sum of successive fundamental polynomials, Ann. of Numer. Math., 2 (1995), 305-309.

    MATH  MathSciNet  Google Scholar 

  9. D. S. Lubinsky and D. H. Mache, (C, 1) means of orthonormal expansions for exponential weights, J. Approx. Theory, 103 (2000), 151-182.

    Article  MATH  MathSciNet  Google Scholar 

  10. H. N. Mhaskar, Introduction to the Theory of Weighted Polynomial Approximation, World Scientific (Singapore-New Jersey-London-Hong Kong, 1996).

    Google Scholar 

  11. H. N. Mhaskar and E. B. Saff, Where does the sup norm of a weighted polynomial live? Constr. Approx. 1 (1985), 71-79.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Szabados, Weighted Lagrange and Hermite-Fejér interpolation on the real line, J. of Inequal. and Appl., 1 (1997), 99-112.

    MATH  MathSciNet  Google Scholar 

  13. J. Szabados and P. Vértesi, Interpolation of Functions, World Sci. Publ. (Singapore-New Jersey-London-Hong Kong, 1990).

    Google Scholar 

  14. G. Szegö, Orthogonal Polynomials, AMS Coll. Publ., Vol. 23 (Providence, 1978).

    Google Scholar 

  15. L. Szili and P. Vértesi, On summability of weighted Lagrange interpolation. I (General weights), Acta Math. Hungar., 101 (2003), 323-344.

    Article  MATH  MathSciNet  Google Scholar 

  16. V. Totik, Weighted Approximation with Varying Weights, Lecture Notes in Mathematics 1569, Springer Verlag (1994).

  17. P. Vértesi, An Erdős-type convergence process in weighted interpolation. I (Freud-type weights, Acta Math. Hungar., 91 (2001), 195-215.

    Article  MATH  MathSciNet  Google Scholar 

  18. P. Vértesi, On the Lebesgue function of weighted Lagrange interpolation. I (Freud-type weights), Constr. Approx., 15 (1999), 355-367.

    Article  MATH  MathSciNet  Google Scholar 

  19. F. Weisz, θ-summation and Hardy spaces, J. Approx. Theory, 107 (2000), 121-142.

    Article  MATH  MathSciNet  Google Scholar 

  20. F. Weisz, Several dimensional θ-summability and Hardy spaces, Math. Nachr., 230 (2001), 159-180.

    Article  MATH  MathSciNet  Google Scholar 

  21. F. Weisz, Marcinkiewicz-θ-summability of Fourier transform, Acta Math. Hungar., 96 (2002), 135-146.

    Article  MATH  MathSciNet  Google Scholar 

  22. F. Weisz, Summability of Multidimensional Fourier Series and Hardy Spaces, Kluwer Academic Publishers (Boston-Dordrecht-London, 2002).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department Of Numerical Analysis, Loránd Eötvös University, H-1117, Budapest, Pázmány P. Sétány I/C

    László Szili

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences Alfréd Rényi Institute of Mathematics, H-1053, Budapest, Reáltanoda U. 13--15

    Péter Vértesi

Authors
  1. László Szili
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Péter Vértesi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Szili, L., Vértesi, P. On summability of weighted Lagrange interpolation. II. Acta Mathematica Hungarica 103, 1–17 (2004). https://doi.org/10.1023/B:AMHU.0000028233.68543.40

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:AMHU.0000028233.68543.40

Search

Navigation