Vector interpretation of the matrix orthogonality on the real line

TL;DR

This paper explores vector orthogonal polynomials as a reinterpretation of matrix orthogonality, focusing on non-symmetric cases, and establishes their orthonormality with respect to complex measures, including applications to approximation and a Markov's theorem.

Contribution

It introduces a vector-based perspective on matrix orthogonality, allowing analysis of non-symmetric systems and proving their orthonormality with complex measures.

Findings

Vector orthogonal polynomials satisfy three-term recurrence relations with non-symmetric matrix coefficients.

The systems are orthonormal with respect to a complex measure.

Includes discussion on Hermite-Padé approximation and a Markov's type theorem.

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-Pad\'e type are also discussed. Finally, a Markov's type theorem is presented.