Bispectrality and Time-Band-Limiting: Matrix valued polynomials

TL;DR

This paper explores matrix-valued bispectrality and its connection to time-band-limiting, providing conditions to construct commuting differential operators that facilitate efficient eigenfunction computation in signal processing.

Contribution

It introduces a general condition for matrix-valued bispectrality and constructs operators commuting with time- and band-limiting operators, extending classical results.

Findings

Established a condition for matrix-valued bispectrality.

Constructed commuting differential operators for time-band-limiting.

Connected bispectrality with integrable systems and signal processing.

Abstract

The subject of time-band-limiting, originating in signal processing, is dominated by the miracle that a naturally appearing integral operator admits a commuting differential one allowing for a numerically efficient way to compute its eigenfunctions. Bispectrality is an effort to dig into the reasons behind this miracle and goes back to joint work with H. Duistermaat. This search has revealed unexpected connections with several parts of mathematics, including integrable systems. Here we consider a matrix valued version of bispectrality and give a general condition under which we can display a constructive and simple way to obtain the commuting differential operator. Furthermore, we build an operator that commutes with both the time-limiting operator and the band-limiting operators.