Time and band limiting for matrix valued functions: an integral and a commuting differential operator

TL;DR

This paper extends the classical time and band limiting problem to matrix-valued functions, demonstrating that the key commuting property between integral and differential operators persists in the matrix setting, enabling potential new signal processing methods.

Contribution

It generalizes the classical scalar time-band limiting problem to matrix-valued functions, establishing the existence of commuting integral and differential operators in this broader context.

Findings

The integral operator commutes with a differential operator for matrix-valued functions.

The classical miracle of commuting operators extends to the matrix case.

This extension opens avenues for advanced matrix-valued signal processing techniques.

Abstract

The problem of recovering a signal of finite duration from a piece of its Fourier transform was solved at Bell Labs in the $1960$'s, by exploiting a "miracle": a certain naturally appearing integral operator commutes with an explicit differential one. Here we show that this same miracle holds in a matrix valued version of the same problem.