Commuting integral and differential operators and the master symmetries of the Korteweg-de Vries equation

TL;DR

This paper explores how the master symmetries of the Korteweg-de Vries equation can extend Slepian's classical results on commuting differential and integral operators, impacting areas like tomography and random matrix theory.

Contribution

It introduces a novel connection between the master symmetries of the KdV equation and commuting operators related to Slepian's work, expanding the theoretical framework.

Findings

Extension of Slepian's results using KdV symmetries

Identification of differential operators commuting with integral operators

Implications for tomography and random matrix theory

Abstract

The singular value decomposition going with many problems in medical imaging, non-destructive testing, geophysics, is of central importance. Unfortunately the effective numerical determination of the singular functions in question is a very ill-posed problem. The best known remedy to this problem goes back to the work of D. Slepian, H.Landau and H. Pollak, Bell Labs 1960-1965. We show that the master symmetries of the Korteweg-de Vries equation give a way to extend the remarkable result of D. Slepian in connection with the Bessel integral kernel and the existence of a differential operator that commutes with the corresponding integral operator. The original results of the Bell Labs group has already played an important role in the study of the limited angle problem in X-ray tomography as well as in Random Matrix theory.