Skip to main content
SpringerLink
Log in

Characteristic Polynomials of Random Banded Hessenberg Matrices and Hermite–Padé Approximation

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

We consider a class of random banded Hessenberg matrices with independent entries having identical distributions along diagonals. The distributions may be different for entries belonging to different diagonals. For a sequence of \(n\times n\) matrices in the class considered, we investigate the asymptotic behavior of their empirical spectral distribution as n tends to infinity.

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

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Angelesco, M.A.: Sur deux extensions des fractions continues algébriques. C. R. Acad. Sci. Paris 168, 262–265 (1919)

    MATH  Google Scholar 

  2. Aptekarev, A., Kaliaguine, V.: Complex rational approximation and difference operators. In: Proceedings of the Third International Conference on Functional Analysis and Approximation Theory (Acquafredda di Maratea, 1996). Rend. Circ. Mat. Palermo (2) Suppl. No. 52, Vol. I (1998), 3–21

  3. Aptekarev, A., Kaliaguine, V., Van Iseghem, J.: The genetic sum’s representation for the moments of a system of Stieltjes functions and its application. Constr. Approx. 16, 487–524 (2000)

  4. Aptekarev, A.I., Kalyagin, V., López Lagomasino, G., Rocha, I.A.: On the limit behavior of recurrence coefficients for multiple orthogonal polynomials. J. Approx. Theory 139, 346–370 (2006)

    Article  MathSciNet  Google Scholar 

  5. Aptekarev, A.I., Kalyagin, V.A., Saff, E.B.: Higher-order three-term recurrences and asymptotics of multiple orthogonal polynomials. Constr. Approx. 30, 175–223 (2009)

    Article  MathSciNet  Google Scholar 

  6. Barrios Rolanía, D., Geronimo, J.S., López Lagomasino, G.: High order recurrence relations, Hermite–Padé approximants, and Nikishin systems. Sb. Math. 209, 385–420 (2018)

    Article  MathSciNet  Google Scholar 

  7. Barrios, D., López, G., Martínez-Finkelshtein, A., Torrano, E.: Finite-dimensional approximations of the resolvent of an infinite band matrix and continued fractions. Sb. Math. 190, 501–519 (1999)

    Article  MathSciNet  Google Scholar 

  8. Dai, D., Ismail, M.E.H., Wang, X.-S.: Doubly infinite Jacobi matrices revisited: resolvent and spectral measure. Adv. Math. 343, 157–192 (2019)

    Article  MathSciNet  Google Scholar 

  9. Damanik, D., Killip, R., Simon, B.: Perturbations of orthogonal polynomials with periodic recursion coefficients. Ann. Math. 171, 1931–2010 (2010)

    Article  MathSciNet  Google Scholar 

  10. Delvaux, S., López, A.: High order three-term recursions, Riemann–Hilbert minors and Nikishin systems on star-like sets. Constr. Approx. 188, 337–398 (2004)

    MATH  Google Scholar 

  11. Delvaux, S., López, A., López Lagomasino, G.: A family of Nikishin systems with periodic recurrence coefficients. Sb. Math. 204, 43–74 (2013)

    Article  MathSciNet  Google Scholar 

  12. Kalyagin, V.A.: Hermite–Padé approximants and spectral analysis of nonsymmetric operators. Sb. Math. 82, 199–216 (1995)

    Article  MathSciNet  Google Scholar 

  13. Leurs, M., Van Assche, W.: Jacobi–Angelesco multiple orthogonal polynomials on an \(r\)-star. Constr. Approx. 51, 353–381 (2020)

    Article  MathSciNet  Google Scholar 

  14. Leurs, M., Van Assche, W.: Laguerre–Angelesco multiple orthogonal polynomials on an \(r\)-star. J. Approx. Theory 250(2020), 105324 (0 p) (2020)

    MathSciNet  MATH  Google Scholar 

  15. López García, A.: Asymptotics of multiple orthogonal polynomials for a system of two measures supported on a starlike set. J. Approx. Theory 163, 1146–1184 (2011)

    Article  MathSciNet  Google Scholar 

  16. López-García, A., Miña-Díaz, E.: Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials. Sb. Math. 209, 1051–1088 (2018)

    Article  MathSciNet  Google Scholar 

  17. López-García, A., Prokhorov, V.A.: On random polynomials generated by a symmetric three-term recurrence relation. J. Differ. Equ. Appl. 25, 1604–1643 (2019)

    Article  MathSciNet  Google Scholar 

  18. Masson, D.R., Repka, J.: Spectral theory of Jacobi matrices in \(l^{2}(\mathbb{Z})\) and the \({{\rm su}}(1,1)\) Lie algebra. SIAM J. Math. Anal. 22, 1131–1146 (1991)

    Article  MathSciNet  Google Scholar 

  19. Nikishin, E.M.: On simultaneous Padé approximants. Math. USSR Sb. 41, 409–425 (1982)

    Article  Google Scholar 

  20. Nikishin, E.M.: Discrete Sturm–Liouville operators and some problems of function theory. Trudy Seminara imeni I.G. Petrovskogo 10 (1984), 3–77; English transl. in J. Math. Sci. 35, 2679–2744 (1986)

  21. Nikishin, E.M., Sorokin, V.N.: Rational Approximations and Orthogonality, Transl. Math. Monographs. Amer. Math. Soc. (92) (1991)

  22. Robert, L., Santiago, L.: Finite sections method for Hessenberg matrices. J. Approx. Theory 123, 68–88 (2003)

    Article  MathSciNet  Google Scholar 

  23. Simon, B.: Szego’s Theorem and Its Descendants: Spectral Theory for \(L^2\) Perturbations of Orthogonal Polynomials. Princeton University Press, Princeton, NJ, M.B. Porter Lectures (2011)

  24. Van Iseghem, J.: Matrix continued fraction for the resolvent function of the band operator. Acta Appl. Math. 61, 351–365 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, 4393 Andromeda Loop North, Orlando, FL, 32816, USA

    A. López-García

  2. Department of Mathematics and Statistics, University of South Alabama, 411 University Boulevard North, Mobile, AL, 36688, USA

    V. A. Prokhorov

Authors
  1. A. López-García
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. A. Prokhorov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. López-García.

Additional information

Communicated by Fabrizio Colombo.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

López-García, A., Prokhorov, V.A. Characteristic Polynomials of Random Banded Hessenberg Matrices and Hermite–Padé Approximation. Complex Anal. Oper. Theory 15, 113 (2021). https://doi.org/10.1007/s11785-021-01162-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11785-021-01162-y

Search

Navigation