Componentwise Products of Totally Non-Negative Matrices Generated by Functions in the Laguerre–Pólya Class
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.
KeywordsSchur-Hadamard product Infinite matrices Aperiodic polynomials Positive pairs Hurwitz-stability Totally positive matrices
- 1.Ahlfors, L.V.: Complex Analysis, 3rd edn. McGraw-Hill Inc., New York (1979)Google Scholar
- 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