Zero asymptotic behaviour for orthogonal matrix polynomials
The main goal of this paper is to extend to the matrix setting some classical results on zero asymptotics of scalar orthogonal polynomials on \(\mathbb R\). This is also a logical continuation of a series of papers of the first author (see the list of references). Consider an \(N \times N\) positive definite matrix of measures \(W\) on \(\mathbb R\) such that the sequence of polynomials \((P_n)_n\), satisfying \[ \int P_n(t) dW(t) P_m^*(t)=\delta_{n,m} I , \quad n, m \geq 0 , \] exists. These polynomials satisfy also a three-term recurrence relation \[ t P_n(t)=A_{n+1} P_{n+1}(t) + B_n P_n(t) +A_n^* P_{n-1}(t) , \quad P_{-1}=0 , \quad P_0 \neq 0 , \] where \(A_n\) are nonsingular and \(B_n\) are hermitian. It is known that the zeros of \(\det (P_n)\) are real and have multiplicity \(\leq N\). Thus, denoting by \(x_{n,k}\) the zeros of \(\det (P_n)\) (with account of their multiplicities), the normalized zero-counting measure for \(P_n\) is \[ \sigma_n={1 \over nN} \sum \delta_{x_{n,k}} . \] The aim of this paper is to study the weak limits of \((\sigma_n)\) in two different situations. First, the authors assume that \(W\) belongs to the matrix Nevai class \(M(A,B)\): \[ \lim_n A_n =A , \quad \lim_n B_n =B . \] When \(A\) is hermitian and nonsingular, they show that \[ \lim_n \sigma_n =\text{tr}\left( {1 \over N} {\mathcal X}_{A, B}\right) , \] where \({\mathcal X}_{A, B}\) is the matrix weight for the Chebyshev matrix polynomials of the first kind, defined by a recurrence relation with constant coefficients \(A, B\). Furthermore, in the particular case when \(A\) is positive definite, the explicit expression for \({\mathcal X}_{A, B}\) is given; the support of \({\mathcal X}_{A, B}\) is the union of \(\leq N\) disjoint bounded nondegenerate intervals on \(\mathbb R\). In the second part of the paper, another classical result is extended. Namely, let \(W'\) be the matrix of Radon-Nikodým derivatives of the entries of \(W\) with respect to \(\text{tr} (W)\), and \(\Delta_k\) is the \(k\)-th principal minor of \(W'\). The authors assume also that the scalar measures \[ {\Delta_k \over \Delta_{k-1}} d(\text{tr} (W)) , \quad k=1, \ldots, N , \] are finite and have the Erdös-Turán property on \([-1,1]\). Then, \[ \lim_n \sigma_n = {1 \over \pi} {dt \over \sqrt{1-t^2}} , \quad t \in [-1,1] . \]
scientific article; zbMATH DE number 1369955
scientific article