Skip to main content
Springer Nature Link
Log in

Asymptotics of orthogonal polynomials and the numerical solution of ill-posed problems

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The paper reviews the impact of modern orthogonal polynomial theory on the analysis of numerical algorithms for ill-posed problems. Of major importance are uniform bounds for orthogonal polynomials on the support of the weight function, the growth of the extremal zeros, and asymptotics of the Christoffel functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. H. Brakhage, On ill-posed problems and the method of conjugate gradients, in:Inverse and Ill-Posed Problems, eds. H. W. Engl and C. W. Groetsch (Academic Press, Boston/New York/London, 1987), pp. 165–175.

    Google Scholar 

  2. T.S. Chihara,An Introduction to Orthogonal Polynomials, (Gordon and Breach, New York, 1978).

    Google Scholar 

  3. T.S. Chihara and M.E.H. Ismail, Orthogonal polynomials suggested by a queueing model, Adv. Appl. Math. 3 (1982) 441–462.

    Article  Google Scholar 

  4. I.S. Gradshteyn and I.M. Ryzhik,Table of Integrals, Series, and Products (Academic Press, New York/San Francisco/London, 1965).

    Google Scholar 

  5. M. Hanke, Accelerated Landweber iterations for the solution of ill-posed equations, Number. Math. 60 (1991) 341–373.

    Article  Google Scholar 

  6. M. Hanke, An ε-free a posteriori stopping rule for certain iterative regularization methods, SIAM J. Numer. Anal. 30 (1993) 1208–1228.

    Article  Google Scholar 

  7. M. Hanke, Regularization of ill-posed problems by conjugate gradient type methods, Habilitationsschrift, Universität Karlsruhe, Karlsruhe (1994).

    Google Scholar 

  8. M. Hanke and H.W. Engl, An optimal stopping rule for thev-methods for solving ill-posed problems using Christoffel functions, J. Approx. Theory 79 (1994) 89–108.

    Article  Google Scholar 

  9. M.E.H. Ismail, The zeros of basic Bessel functions, the functionsJ v+ax (x), and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982) 1–19.

    Article  Google Scholar 

  10. K.G. Ivanov and V. Totik, Fast decreasing polynomials, Constr. Approx. 6 (1990) 1–20.

    Article  Google Scholar 

  11. A.L. Levin and D.S. Lubinsky,Christoffel Functions and Orthogonal Polynomials for Exponential Weights on [−1, 1], Mem. Amer. Math. Soc., No. 535 (Amer. Math. Soc., Providence, RI, 1994).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Praktische Mathematik, Universität Karlsruhe, D-76128, Karlsruhe, Germany

    Martin Hanke

Authors
  1. Martin Hanke
    View author publications

    You can also search for this author inPubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hanke, M. Asymptotics of orthogonal polynomials and the numerical solution of ill-posed problems. Numer Algor 11, 203–213 (1996). https://doi.org/10.1007/BF02142497

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02142497

Keywords