$N\times N$ Matrix Time-/Band- Limiting examples

Abstract

Time- and band-limiting in the context of orthogonal polynomials has been studied since the 1980's. It involves finding differential or difference operators with special commutative properties. More recently this topic has been generalized to the cases were the orthogonal polynomials are matrix-valued. Leading to differential or difference operators with matrix coefficients. So far the only explicit examples of such operators were $2\times 2$ matrices. In this paper we give a number of $N\times N$ examples and a counterexample to illustrate the role that strong Pearson equations can play in finding such examples.