Skip to main content
SpringerLink
Log in

Constrained energy problems with applications to orthogonal polynomials of a discrete variable

  • Published:
Journal d’Analyse Mathematique Aims and scope

Abstract

Given a positive measure Σ with gs > 1, we write Με ℳΣ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight functionw, we prove existence and uniqueness of a measure λΣ w that minimizes the weighted logarithmic energy over the class ℳΣ. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure λΣ w As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk 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. Dragnev,Constrained energy problems for logarithmic potentials, Ph.D. Thesis, University of South Florida, Tampa (1997).

    Google Scholar 

  2. P. Dragnev and E. B. Saff,A problem in potential theory and zero asymptotics of Krawtchouk polynomials (submitted).

  3. P. Erdös and P. Turán,On interpolation III, Ann. of Math.41(1940), 510–553.

    Article  MathSciNet  Google Scholar 

  4. O. Frostman,Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions, Thesis, Meddel, Lunds Univ. Mat. Sem.3 (1935), 1–118.

    Google Scholar 

  5. E. Hille,Analytic Function Theory, Vol. 2, Ginn & Co., Boston, 1962.

    MATH  Google Scholar 

  6. N. S. Landkof,Foundations of Modern Potential Theory, Springer-Verlag, New York, 1972. 259

    MATH  Google Scholar 

  7. V. Levenshtein,Krawtchouk polynomials and universal bounds for codes and design in Hamming spaces, IEEE Trans. Inform. Theory41 (1995), 1303–1321.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. D. Landau and E. M. Lifshitz,Electrodynamics of Continuous Media, 2nd ed., inCourse of Theoretical Physics, Vol. 8, Pergamon Press, Oxford, 1984.

    Google Scholar 

  9. R. J. McEliece, E. R. Rodemich, H. C. Rumsey and L. R. Welch,New upper bounds on the rate of a code via the Delsarte—MacWilliams inequalities, IEEE Trans. Inform. TheoryI-23 (1977), 157–166.

    Article  MathSciNet  Google Scholar 

  10. H. N. Mhaskar and E. B. Saff,Weighted analogues of capacity, transfinite diameter, and Chebyshev constant, Constr. Approx.8 (1992), 105–124.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. A. Rakhmanov,Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable, Mat. Sb.187 (8) (1996), 109–124 (in Russian) = Sbornik:Mathematics187 (8) (1996), 1213–1228 (Engl. transi.).

    MathSciNet  Google Scholar 

  12. E. B. Saff and V. Totik,Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften, Vol. 316, Springer-Verlag, Heidelberg, 1997.

    Google Scholar 

  13. G. Szegö,Orthogonal Polynomials, Vol. 23of Colloquium Publications, Amer. Math. Soc., Providence, R.I., 1975.

    Google Scholar 

  14. M. Tsuji,Potential Theory in Modem Function Theory, Maruzen, Tokyo, 1959.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of South Florida, 33620, Tampa, FL, USA

    P. D. Dragnev

  2. Institute for Constructive Mathematics Department of Mathematics, University of South Florida, 33620, Tampa, FL, USA

    E. B. Saff

Authors
  1. P. D. Dragnev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. B. Saff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. D. Dragnev.

Additional information

The research done by this author is in partial fulfillment of the Ph.D. requirements at the University of South Florida.

The research done by this author was supported, in part, by U.S. National Science Foundation under grant DMS-9501130.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dragnev, P.D., Saff, E.B. Constrained energy problems with applications to orthogonal polynomials of a discrete variable. J. Anal. Math. 72, 223–259 (1997). https://doi.org/10.1007/BF02843160

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02843160

Keywords

Search

Navigation