The prolate spheroidal phenomena and bispectrality

TL;DR

This paper explores the bispectral phenomena in integral operators derived from algebraic structures, revealing their commuting properties and applications in mathematical physics and random matrix theory.

Contribution

It introduces large classes of integral operators from bispectral algebras that exhibit the prolate spheroidal phenomena, extending known examples from special points in Grassmannians.

Findings

Integral operators from bispectral algebras have commuting differential operators.

Large classes of such operators are derived from rank 1 and 2 bispectral algebras.

Connections to time-band limiting and random matrix theory are established.

Abstract

Landau, Pollak, Slepian, and Tracy, Widom discovered that certain integral operators with so called Bessel and Airy kernels possess commuting differential operators and found important applications of this phenomena in time-band limiting and random matrix theory. In this paper we announce that very large classes of integral operators derived from bispectral algebras of rank 1 and 2 (parametrized by lagrangian grassmannians of infinitely large size) have this property. The above examples come from special points in these grassmannians.