On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

A favard type theorem for Hurwitz polynomials

“…In this paper, we crucially use the explicit relation between a Hurwitz polynomial and a member of a family of orthogonal polynomials and its second kind polynomial. This relation, to the knowledge of the authors, was first presented in [6]. The reason why the latter relation was not noted by previous authors lies mainly in two facts.…”

“…Our first motivation is to develop the connection between the Hurwitz polynomials [21,25] and orthogonal polynomials [4,27], which was started in [6]; see also [7] and [8]. For this purpose, let us recall the well-known Favard theorem, which basically states that if a sequence of polynomials (p n (x)) ∞ n=0 , each of them of degree n, satisfies a three-term recurrence relation, then (p n (x)) ∞ n=0 is a sequence of orthogonal polynomials with respect to some distribution function; see e.g.…”

“…Recently an explicit relation between orthogonal polynomials, their second kind polynomials and Hurwitz polynomials was found by the author in [6].…”

“…In [6], it was proved that every Hurwitz polynomial can be written in terms of a member of a family of orthogonal polynomials (p k,j ) Ξ j=0 , k = 1, 2 on [0, ∞) and their second kind polynomials (q k,j ) Ξ j=0 , k = 1, 2:…”

Robust Stability of Hurwitz Polynomials Associated with Modified Classical Weights

“…In [14,15], the authors establish the existence of a one-to-one correspondence between a Hurwitz polynomial and a (finite) sequence of orthogonal polynomials. Furthermore, both topics have well-known connections with Padé approximants [16], the moment problem theory [2,17,18], continued fractions [19][20][21], total positivity of matrices [22], positive functions [23] and the stability and robust stabilization of continuous linear systems [24,25]. More precisely, the next results show how to compute a sequence of Hurwitz polynomials from a SMOP.…”