Matrix Bispectrality and Noncommutative Algebras: beyond the prolate spheroidals

TL;DR

This paper explores a noncommutative extension of the bispectral problem by analyzing matrix-valued operators and algebras, aiming to deepen understanding of bispectral phenomena beyond classical scalar cases.

Contribution

It introduces a noncommutative framework for the bispectral problem using matrix-valued objects and investigates bispectral algebras and their presentations.

Findings

Development of a noncommutative bispectral framework

Analysis of bispectral algebras as tools for understanding bispectral triples

Extension of classical bispectral phenomena to matrix-valued settings

Abstract

The bispectral problem is motivated by an effort to understand and extend a remarkable phenomenon in Fourier analysis on the real line: the operator of time-and-band limiting is an integral operator admitting a second-order differential operator with a simple spectrum in its commutator. In this article, we discuss a noncommutative version of the bispectral problem, obtained by allowing all objects in the original formulation to be matrix-valued. Deep attention is given to bispectral algebras and their presentations as a tool to get information about bispectral triples.