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

TL;DR

This paper extends the study of matrix-valued time- and band-limiting operators from 2x2 cases to general NxN cases, providing new examples and highlighting the role of Pearson equations.

Contribution

It introduces new explicit NxN matrix examples of time- and band-limiting operators and discusses the significance of strong Pearson equations in their construction.

Findings

Provided new NxN matrix examples of limiting operators.

Presented a counterexample illustrating the role of Pearson equations.

Extended the theory from 2x2 to general NxN matrices.

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.