ORTHOGONAL MATRIX POLYNOMIALS AND HIGHER-ORDER RECURRENCE RELATIONS

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 x 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.