1993/10/05 by Antonio J. Durán, Walter Van Assche, Durán, Antonio J. +1 · 3 citations
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.math/9310220
openalex publication_date 1993/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. In this paper 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.