Orthogonal matrix polynomials and higher-order recurrence relations

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Abstract

It is well known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel measure on the real line. We extend this result and show that every system of polynomials satisfying some (2N+1)-term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are N × N matrices. We apply this result to polynomials orthogonal with respect to a discrete Sobolev inner product and other inner products in the linear space of polynomials. As an application we give a short proof of Krein's characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points. © 1995.

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Durán, A. J., & Van Assche, W. (1995). Orthogonal matrix polynomials and higher-order recurrence relations. Linear Algebra and Its Applications, 219(C), 261–280. https://doi.org/10.1016/0024-3795(93)00218-O

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