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.
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References
Ahlfors, L.V.: Complex Analysis, 3rd edn. McGraw-Hill Inc., New York (1979)
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)
Asner, B.A.: On the total nonnegativity of the Hurwitz matrix. SIAM J. Appl. Math. 18, 407–414 (1970)
Boas, R.P.: Entire Functions. Academic Press Inc., New York (1954)
Dyachenko, A.: Total nonnegativity of infinite Hurwitz matrices of entire and meromorphic functions. Complex Anal. Oper. Theory 8(5), 1097–1127 (2014)
Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton University Press, Princeton (2011)
Gantmacher, F.: Matrizentheorie. Springer, Berlin (1986)
Garloff, J., Wagner, D.G.: Hadamard products of stable polynomials are stable. J. Math. Anal. Appl. 202, 797–809 (1996)
Holtz, O., Tyaglov, M.: Structured matrices, continued fractions, and root localization of polynomials. SIAM Rev. 54(3), 421–509 (2012)
Hurwitz, A.: Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. Math. Ann. 46, 273–284 (1895)
Kemperman, J.H.B.: A Hurwitz matrix is totally positive. SIAM J. Math. Anal. 13, 331–341 (1982)
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)
Levin, B.J.: Distribution of Zeros of Entire Functions, Translation of Mathematical Monographs, vol. 5 (AMS, Providence, Rhode Island, 1980), 2nd, revised edition
Pinkus, A.: Totally Positive Matrices, 2nd edn. AMS, Rhode Island (1980)
Pólya, G., Schur, I.: Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144, 89–113 (1914)
Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Oxford University Press, Oxford (2002)
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Batra, P. (2017). Componentwise Products of Totally Non-Negative Matrices Generated by Functions in the Laguerre–Pólya Class. In: Bebiano, N. (eds) Applied and Computational Matrix Analysis. MAT-TRIAD 2015. Springer Proceedings in Mathematics & Statistics, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-49984-0_11
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DOI: https://doi.org/10.1007/978-3-319-49984-0_11
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