Skip to main content
SpringerLink
Log in

Pointwise bounds of orthogonal expansions on the real line via weighted Hilbert transforms

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

Abstract

We study pointwise bounds of orthogonal expansions on the real line for a class of exponential weights of smooth polynomial decay at infinity. As a consequence of our main results, we establish pointwise bounds for weighted Hilbert transforms which are of independent interest.

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. R. Askey and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, J. Math. 87 (1965) 695–708.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. Bojanic and H.N. Mhaskar, A rate of convergence theorem for expansions in Hermite polynomials, Manuscript (1988).

  3. S.B. Damelin, The Lebesgue constant of Lagrange interpolation for Erdős weights, J. Approx. Theory 94(2) (1998) 235–262.

    Article  MATH  MathSciNet  Google Scholar 

  4. S.B. Damelin, The Hilbert transform and orthonormal expansions for exponential weights, in: Approximation Theory X: Abstract and Classical Analysis, eds. Chui, Schumaker and Stoekler (Vanderbilt Univ. Press, Nashville, TN, 2002) pp. 117–135.

    Google Scholar 

  5. S.B. Damelin, Asymptotics of recurrence coefficients for exponential weights, Magnus's method revisited, Math. Comp. 73 (2004) 191–209.

    Article  MATH  MathSciNet  Google Scholar 

  6. S.B. Damelin, Marcinkiewicz–Zygmund inequalities and the numerical approximation of singular integrals for exponential weights: Methods, results and open problems, some new, some old, J. Complexity 19 (2003) 406–415.

    Article  MATH  MathSciNet  Google Scholar 

  7. S.B. Damelin, On upper bounds for Lebesgue constants via orthonormal expansions, Manuscript.

  8. S.B. Damelin and K. Diethelm, Boundedness and pointwise approximation of the weighted Hilbert transform on the real line, Numer. Funct. Anal. Optim. 22(1/2) (2001) 13–54.

    Article  MATH  MathSciNet  Google Scholar 

  9. S.B. Damelin, H. Jung and K.H. Kwon, Converse Marcinkiewicz–Zygmund inequalities on the real line with applications to mean convergence of Lagrange interpolation, Analysis 22 (2002) 33–55.

    MATH  MathSciNet  Google Scholar 

  10. P. Deift, T. Kriecherbauer, K. McLaughlin, S. Venakides and X. Zhou, Strong asymptotics of orthonormal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999) 1491–1552.

    Article  MATH  MathSciNet  Google Scholar 

  11. G. Freud, Orthonormal Polynomials (Akadémiai Kiadó, Budapest, 1971).

    Google Scholar 

  12. G. Freud, Extension of the Dirichlet–Jordan cirterion to a general class of orthogonal expansions, Acta Math. Acad. Sci. Hung. 25 (1974) 109–122.

    Article  MATH  MathSciNet  Google Scholar 

  13. G. Freud, On the greatest zero of orthonormal polynomials, J. Approx. Theory 46 (1986) 15–23.

    Article  Google Scholar 

  14. S.W. Jha and D.S. Lubinsky, Necessary and sufficient conditions for mean convergence of orthonormal expansions for Freud weights, Constr. Approx. 11 (1995) 331–363.

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Kriecherbauer and K. McLaughlin, Strong asymptotics of polynomials orthonormal with respect to Freud weights, Int. Math. Res. Notices 6 (1999) 299–333.

    Article  MATH  MathSciNet  Google Scholar 

  16. D.S. Lubinsky, A survey of mean convergence of orthogonal expansions, in: Proc. of the 2nd Conf. on Function Spaces, Edwardsville, IL, ed. K. Jarosz (Marcel Dekker, New York, 1995) pp. 281–310.

    Google Scholar 

  17. A.L. Levin and D.S. Lubinsky, L ∞ Markov and Bernstein inequalities for Freud weights, SIAM J. Math. Anal. 21 (1990) 1065–1082.

    Article  MATH  MathSciNet  Google Scholar 

  18. A.L. Levin and D.S. Lubinsky, Orthonormal Polynomials for Exponential Weights (Springer-Verlag, Berlin, 2001).

    Google Scholar 

  19. H.P. Mashele, Extension of the Dirichlet–Jordan convergence criterion for exponential weights, Manuscript.

  20. H.N. Mhaskar, Extensions of Dirichlet–Jordan convergence criterion to a general class of orthogonal polynomial expansions, J. Approx. Theory 42 (1984) 138–148.

    Article  MATH  MathSciNet  Google Scholar 

  21. H.N. Mhaskar, A rate of convergence theorem for expansions in Freud polynomials, J. Approx. Theory 55 (1988) 150–171.

    Article  MATH  MathSciNet  Google Scholar 

  22. H.N. Mhaskar, Introduction to the Theory of Weighted Polynomial Approximation (World Scientific, Singapore, 1996).

    MATH  Google Scholar 

  23. H.N. Mhaskar and Y. Xu, Mean convergence of expansions in Freud type orthogonal polynomials, SIAM J. Math. Anal. 22 (1991) 847–855.

    Article  MATH  MathSciNet  Google Scholar 

  24. B. Muckenhoupt, Mean convergence of Hermite and Laguerre series I, Trans. Amer. Math. Soc. 147 (1970) 419–431.

    Article  MATH  Google Scholar 

  25. B. Muckenhoupt, Mean convergence of Hermite and Laguerre series II, Trans. Amer. Math. Soc. 147 (1970) 433–460.

    Article  MathSciNet  Google Scholar 

  26. Y.G. Shi, Bounds and inequalities for general orthogonal polynomials on finite intervals, J. Approx. Theory 73 (1993) 303–333.

    Article  MATH  MathSciNet  Google Scholar 

  27. J. Szabados, Weighted Lagrange and Hermite–Fejér interpolation on the real line, J. Inequal. Appl. 1 (1997) 99–123.

    Article  MATH  MathSciNet  Google Scholar 

  28. E.C. Titchmarsh, The Theory of Functions (Oxford Univ. Press, Oxford).

  29. Y. Xu, Mean convergence of generalized Jacobi series and interpolating polynomials, J. Approx. Theory 72 (1993) 237–251.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Georgia Southern University, PO Box 8093, Statesboro, GA, 30460, USA

    S. B. Damelin

Authors
  1. S. B. Damelin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. B. Damelin.

Additional information

Communicated by L. Reichel

Mathematics subject classifications (2000)

41A10, 42C05

Rights and permissions

Reprints and permissions

About this article

Cite this article

Damelin, S.B. Pointwise bounds of orthogonal expansions on the real line via weighted Hilbert transforms. Adv Comput Math 26, 451–471 (2007). https://doi.org/10.1007/s10444-005-7472-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-005-7472-7

Keywords

Search

Navigation