Time and Band Limiting for Matrix Valued Functions, an Example

TL;DR

This paper extends the concept of time and band limiting to matrix-valued functions and demonstrates that a bispectral property leads to a commuting local operator in a non-commutative setting, generalizing classical results.

Contribution

It introduces a non-commutative example linking bispectral properties to commuting operators in matrix-valued orthogonal polynomials and spherical functions.

Findings

First non-commutative example of this kind

Bispectral property implies a commuting local operator

Generalizes classical prolate spheroidal wave operator

Abstract

The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of "time and band limiting" admits a commuting local operator. This is a noncommutative analog of the famous prolate spheroidal wave operator.