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Componentwise Products of Totally Non-Negative Matrices Generated by Functions in the Laguerre–Pólya Class

  • Prashant Batra
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 192)

Abstract

In connection with the characterisation of real polynomials which have exclusively negative zeros Holtz and Tyaglov exposed in 2012 a new, totally non-negative, infinite matrix. This matrix resembles the matrices considered in the stability problem, and was called a matrix of “Hurwitz-type”. No precise connection to the Hurwitz matrices of the stability problem or structural properties could be established. We identify those matrices as limits of Hurwitz matrices generated by Hurwitz-stable polynomials . This allows to give a new and concise proof of the Holtz–Tyaglov characterisation as we connect it here to the classical theorem of Aissen, Edrei, Schoenberg and Whitney. Our approach naturally extends to entire functions in the Laguerre–Pólya class which have exclusively non-negative Taylor coefficients. Results on Hurwitz-stable polynomials are employed to show that certain positive pairs of real functions in the Laguerre–Pólya class generate totally non-negative matrices. Finally, we give the first composition result on the structured, infinite matrices considered: We show that the componentwise product of any of the considered infinite matrices is totally non-negative.

Keywords

Schur-Hadamard product Infinite matrices Aperiodic polynomials Positive pairs Hurwitz-stability Totally positive matrices 

References

  1. 1.
    Ahlfors, L.V.: Complex Analysis, 3rd edn. McGraw-Hill Inc., New York (1979)Google Scholar
  2. 2.
    Aissen, M., Edrei, A., Schoenberg, I.J., Whitney, A.: On the generating functions of totally positive sequences. Proc. Natl. Acad. Sci. U. S. A. 37, 303–307 (1951)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Asner, B.A.: On the total nonnegativity of the Hurwitz matrix. SIAM J. Appl. Math. 18, 407–414 (1970)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boas, R.P.: Entire Functions. Academic Press Inc., New York (1954)MATHGoogle Scholar
  5. 5.
    Dyachenko, A.: Total nonnegativity of infinite Hurwitz matrices of entire and meromorphic functions. Complex Anal. Oper. Theory 8(5), 1097–1127 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton University Press, Princeton (2011)CrossRefMATHGoogle Scholar
  7. 7.
    Gantmacher, F.: Matrizentheorie. Springer, Berlin (1986)CrossRefGoogle Scholar
  8. 8.
    Garloff, J., Wagner, D.G.: Hadamard products of stable polynomials are stable. J. Math. Anal. Appl. 202, 797–809 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Holtz, O., Tyaglov, M.: Structured matrices, continued fractions, and root localization of polynomials. SIAM Rev. 54(3), 421–509 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hurwitz, A.: Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. Math. Ann. 46, 273–284 (1895)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kemperman, J.H.B.: A Hurwitz matrix is totally positive. SIAM J. Math. Anal. 13, 331–341 (1982)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Krein, M.G., Naimark, M.A.: The method of symmetric and hermitian forms in the theory of the separation of the roots of algebraic equations. Linear Multilinear Algebra 10, 265–308 (1981)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Levin, B.J.: Distribution of Zeros of Entire Functions, Translation of Mathematical Monographs, vol. 5 (AMS, Providence, Rhode Island, 1980), 2nd, revised editionGoogle Scholar
  14. 14.
    Pinkus, A.: Totally Positive Matrices, 2nd edn. AMS, Rhode Island (1980)MATHGoogle Scholar
  15. 15.
    Pólya, G., Schur, I.: Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144, 89–113 (1914)MathSciNetMATHGoogle Scholar
  16. 16.
    Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Oxford University Press, Oxford (2002)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany

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