Vector Interpretation of the Matrix Orthogonality on the Real Line

Abstract

In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients that do not obey to any type of symmetry. In this sense the vectorial reinterpretation allows us to study a non-symmetric case of the matrix orthogonality. We also prove that our systems of polynomials are indeed orthonormal with respect to a complex measure of orthogonality. Approximation problems of Hermite-Padé type are also discussed. Finally, a Markov's type theorem is presented.

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

“…M , we have that the last relation is equivalent to (18). Now, we prove that (c) ⇒ (d) remember that F can be written…”

“…Now, consider the space of vector of polynomials P 2 = P j , j ∈ N , where P j = x 2j P 0 with P 0 = 1 x T , and the space M 2×2 (C) of 2×2-matrices with complex entries. It is well known (see [18]) that there exist a vector of linear functionals U = u 1 u 2 T defined in (P 2 ) * , the linear space of vector linear functionals, here called dual space, acting in P 2 over M 2×2 (C) such that…”

“…In [18], necessary and sufficient conditions for the quasi-definiteness of U, i.e., for the existence of a vector sequence of polynomials leftorthogonal with respect to the vector of linear functionals U, were obtained. These sequences of polynomials satisfy non-symmetric three term recurrence relations.…”

“…Moreover, the coefficients of the three term recurrence relation can be expressed in terms of a vector of linear functionals U or in terms of a matrix measure associated with {V m } or {G n } (cf. [18]). Furthermore, left and right vector orthogonality is connected with right and left matrix orthogonality as will see bellow.…”

“…[18]). The matrix sequence {G n } and the vector sequence of polynomials {B m } are bi-orthogonal with respect to U if, and only if, the sequence of matrix orthogonal polynomials {G n } and {V m } are bi-orthogonal with respect to F, i.e.,…”