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The Smallest Eigenvalue of Hankel Matrices

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Abstract

Let ℋ N =(s n+m),0≤n,m≤N, denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behavior of the smallest eigenvalue λ N of ℋ N . It is proven that λ N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of λ N can be arbitrarily slow or arbitrarily fast in a sense made precise below. In the indeterminate case, where λ N is known to be bounded below by a strictly positive constant, we prove that the limit of the nth smallest eigenvalue of ℋ N for N→∞ tends rapidly to infinity with n. The special case of the Stieltjes–Wigert polynomials is discussed.

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References

  1. Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver and Boyd, Edinburgh (1965), English translation

    MATH  Google Scholar 

  2. Beckermann, B.: The condition number of real Vandermonde, Krylov and positive definite Hankel matrices. Numer. Math. 85, 553–577 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berg, C.: Fibonacci numbers and orthogonal polynomials. arXiv:math.NT/0609283

  4. Berg, C., Chen, Y., Ismail, M.E.H.: Small eigenvalues of large Hankel matrices: the indeterminate case. Math. Scand. 91, 67–81 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Berg, C., Christensen, J.P.R.: Density questions in the classical theory of moments. Ann. Inst. Fourier Grenoble 31(3), 99–114 (1981)

    MathSciNet  MATH  Google Scholar 

  6. Berg, C., Durán, A.J.: The index of determinacy for measures and the ℓ 2-norm of orthogonal polynomials. Trans. Am. Math. Soc. 347, 2795–2811 (1995)

    Article  MATH  Google Scholar 

  7. Berg, C., Durán, A.J.: Orthogonal polynomials and analytic functions associated to positive definite matrices. J. Math. Anal. Appl. 315, 54–67 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Y., Lawrence, N.D.: Small eigenvalues of large Hankel matrices. J. Phys. A 32, 7305–7315 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Chen, Y., Lubinsky, D.S.: Smallest eigenvalues of Hankel matrices for exponential weights. J. Math. Anal. Appl. 293, 476–495 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chihara, T.: Chain sequences and orthogonal polynomials. Trans. Am. Math. Soc. 104, 1–16 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  11. Christiansen, J.S.: The moment problem associated with the Stieltjes–Wigert polynomials. J. Math. Anal. Appl. 277, 218–245 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Collar, A.R.: On the Reciprocation of Certain Matrices. Proc. R. Soc. Edinb. 59, 195–206 (1939)

    MathSciNet  Google Scholar 

  13. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990), second edition 2004

    MATH  Google Scholar 

  14. Lubinsky, D.S.: Condition numbers of Hankel matrices for exponential weights. J. Math. Anal. Appl. 314, 266–285 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Shohat, J., Tamarkin, J.D.: The Problem of Moments. American Mathematical Society, Providence (1950), revised edition

    MATH  Google Scholar 

  16. Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Simon, B.: The Christoffel–Darboux kernel. In: Perspectives in PDE, Harmonic Analysis and Applications. A Volume in Honor of V.G. Maz’ya’s 70th Birthday. Proceedings of Symposia in Pure Mathematics, vol. 79, pp. 295–335. Birkhäuser, Basel (2008)

    Google Scholar 

  18. Stieltjes, T.J.: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, 1–122 (1894); 9, 5–47 (1895). English translation in Thomas Jan Stieltjes, Collected Papers, vol. II, pp. 609–745. Springer, Berlin, Heidelberg, New York, 1993

    MathSciNet  Google Scholar 

  19. Szegő, G.: On some Hermitian forms associated with two given curves of the complex plane. Trans. Am. Math. Soc. 40, 450–461 (1936). In: Collected Papers (vol. 2), pp. 666–678. Birkhäuser, Boston, Basel, Stuttgart, 1982

    Google Scholar 

  20. Szegő, G.: Orthogonal Polynomials, 4th edn. Colloquium Publications, vol. 23. Am. Math. Soc., Rhode Island (1975)

    Google Scholar 

  21. Szwarc, R.: A lower bound for orthogonal polynomials with an application to polynomial hypergroups. J. Approx. Theory 81, 145–150 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Szwarc, R.: Absolute continuity of certain unbounded Jacobi matrices. In: Buhmann, M.D., Mache, D.H. (eds.) Advanced Problems in Constr. Approx. International Series of Numerical Mathematics, vol. 142, pp. 255–262. Birkhäuser, Basel (2003)

    Google Scholar 

  23. Widom, H., Wilf, H.S.: Small eigenvalues of large Hankel matrices. Proc. Am. Math. Soc. 17, 338–344 (1966)

    Article  MathSciNet  Google Scholar 

  24. Wigert, S.: Sur les polynomes orthogonaux et l’approximation des fonctions continues. Ark. Mat. Astron. Fys. 17(18), 15 (1923)

    Google Scholar 

  25. Wilf, H.S.: Finite Sections of Some Classical Inequalities. Springer, Berlin, Heidelberg, New York (1970)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Copenhagen, Universitetsparken 5, Copenhagen, 2100, Denmark

    Christian Berg

  2. Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Ryszard Szwarc

  3. Institute of Mathematics and Computer Science, University of Opole, ul. Oleska 48, 45-052, Opole, Poland

    Ryszard Szwarc

Authors
  1. Christian Berg
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  2. Ryszard Szwarc
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Correspondence to Christian Berg.

Additional information

Communicated by Doron S. Lubinsky.

The present work was initiated while the first author was visiting University of Wrocław under a grant by the HANAP project mentioned under the second author. The first author has been supported by grant 272-07-0321 from the Danish Research Council for Nature and the Universe.

The second author was supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by MNiSW Grant N201 054 32/4285.

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Berg, C., Szwarc, R. The Smallest Eigenvalue of Hankel Matrices. Constr Approx 34, 107–133 (2011). https://doi.org/10.1007/s00365-010-9109-4

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  • DOI: https://doi.org/10.1007/s00365-010-9109-4

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