Skip to main content
SpringerLink
Log in

Powers of the Dirichlet kernel with respect to orthogonal polynomials and related operators

  • Published:
Acta Scientiarum Mathematicarum Aims and scope Submit manuscript

Abstract

Let {Pn}∞n=0 be an orthogonal polynomial sequence on the real line with respect to a probability measure µ with compact and infinite support and

$${D_N} = \sum\nolimits_{n = 0}^N {{P_n}{h_n}}$$

the Nth element of the Dirichlet kernel, where hn = (∫ Pn2dµ)-1. We are investigating the rth integer power DrN and prove for special orthogonal polynomials that in the case r ∈ ℕ {1} the sequence {DNr∞N=0{ gives rise to an approximate identity. This applies for example for Jacobi polynomials.

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. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser Verlag, Basel, 1971.

    Book  Google Scholar 

  2. T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978.

    MATH  Google Scholar 

  3. G. Faber, Über die interpolatorische Darstellung stetiger Funktionen, Jahresber. Deutsch. Math. Verein., 23 (1914), 192–210.

    MATH  Google Scholar 

  4. G. Gasper, Linearization of the product of Jacobi polynomials I, Canad. J. Math., 22 (1970), 171–175.

    Article  MathSciNet  Google Scholar 

  5. G. Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math., 95 (1972), 261–280.

    Article  MathSciNet  Google Scholar 

  6. T. P. Laine, The product formula and convolution structure for the generalized Chebyshev polynomials, SIAM J. Math. Anal., 11 (1980), 133–146.

    Article  MathSciNet  Google Scholar 

  7. R. Lasser, Orthogonal polynomials and hypergroups, Rend. Mat., 3 (1983), 185–209.

    MathSciNet  MATH  Google Scholar 

  8. R. Lasser and J. Obermaier, On the convergence of weighted Fourier expansions, Acta. Sci. Math. (Szeged), 61 (1995), 345–355.

    MathSciNet  MATH  Google Scholar 

  9. J. Obermaier, A continuous function space with a Faber basis, J. Approx. Theory, 125 (2003), 303–312.

    Article  MathSciNet  Google Scholar 

  10. J. Obermaier and R. Szwarc, Orthogonal polynomials of discrete variable and boundedness of Dirichlet kernel, Constr. Approx., 27 (2008), 1–13.

    Article  MathSciNet  Google Scholar 

  11. G. Szegö, Orthogonal Polynomials, Amer. Math. Soc, Providence, RI, 1959.

    MATH  Google Scholar 

  12. R. Szwarc, Orthogonal polynomials and Banach algebras, Advances in the Theory of special functions and Orthogonal olynomials, Nova Science Publishers Vol. 2, 2005, 103–139.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Helmholtz Zentrum München, German Research Center for Environmental Health, Institute of Computational Biology, Ingolstädter Landstrasse 1, D-85764, Neuherberg, Germany

    Josef Obermaier

Authors
  1. Josef Obermaier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Josef Obermaier.

Additional information

Communicated by V. Totik

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Obermaier, J. Powers of the Dirichlet kernel with respect to orthogonal polynomials and related operators. ActaSci.Math. 83, 539–549 (2017). https://doi.org/10.14232/actasm-016-072-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.14232/actasm-016-072-z

AMS Subject Classification

Key words and phrases

Search

Navigation