Total positivity of Hadamard products

Abstract

The Hadamard product of two totally positive Toeplitz matrices M and N need not be totally positive. When only finitely many diagonals of M and of N are non-zero, preservation of total positivity by Hadamard product is essentially a theorem of Maló. Here we establish another sufficient condition for the preservation of total positivity: if both M and N are totally positive lower triangular Toeplitz matrices such that the value on the nth diagonal is a polynomial function of n, then the Hadamard product M · N is totally positive. We use the characterization of the generating functions of Pólya frequency sequences given by Aissen, Edrei, Schoenberg, and Whitney, and in the course of the proof we extend the concept of Sturm sequences, and develop some other results on polynomials with only real roots which are of independent interest.