A new commutativity property of exceptional orthogonal polynomials

TL;DR

This paper demonstrates that a specific commutativity property related to time-and-band limiting, previously known in classical contexts, also applies to exceptional orthogonal polynomials, with potential computational benefits.

Contribution

The authors show that the time-and-band limiting commutative property extends to exceptional orthogonal polynomials, providing new examples and numerical illustrations.

Findings

The property holds for three new examples of exceptional orthogonal polynomials.

Local operators with simple spectrum commute with global operators in these cases.

Numerical evidence suggests advantages of using local operators in computations.

Abstract

We exhibit three examples showing that the "time-and-band limiting" commutative property found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy and H. Widom in Random matrix theory, holds for exceptional orthogonal polynomials. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones. We illustrate numerically the advantage of having such a local operator.