Time and band limiting for exceptional polynomials

TL;DR

This paper extends the classical time-and-band limiting theory to exceptional orthogonal polynomials, establishing the existence of commuting differential operators using Fourier algebras, with applications to Hermite and Laguerre types.

Contribution

It provides a general framework for constructing commuting differential operators for exceptional orthogonal polynomials using Fourier algebras, beyond traditional bispectrality.

Findings

Existence of commuting differential operators for exceptional polynomials.

Explicit examples for Hermite and Laguerre type polynomials.

Representation of operators in Perline's form.

Abstract

The "time-and-band limiting" commutative property was 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. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones. Here we give a general result that insures the existence of a commuting differential operator for a given family of exceptional orthogonal polynomials satisfying the "bispectral property". As a main tool we go beyond bispectrality and make use of the notion of Fourier Algebras associated to the given sequence of exceptional polynomials. We illustrate this result with two examples, of Hermite and Laguerre type, exhibiting also a nice Perline's form for the commuting differential operator.