Bispectrality and Time-Band-Limiting: Matrix valued polynomials

Abstract

The subject of time-band-limiting, originating in signal processing, is dominated by the miracle that a naturallyappearing integral operator admits a commuting dierential one, allowing for a numerically ecient way to compute itseigenfunctions. Bispectrality is an eort 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 integrablesystems. Here we consider a matrix-valued version of bispectrality and give a general condition under which we candisplay a constructive and simple way to obtain the commuting dierential operator. Furthermore, we build an operatorthat commutes with both the time-limiting operator and the band-limiting operators.