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

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.

References in corpus (10)

• Level-Spacing Distributions and the Airy Kernel
• Fredholm Determinants, Differential Equations and Matrix Models
• Level-Spacing Distributions and the Bessel Kernel
• Spatiospectral concentration on a sphere
• Spherical Slepian functions and the polar gap in geodesy
• Quantum Spin Systems and Supersymmetric Gauge Theories, I
• Reflective prolate-spheroidal operators and the KP/KdV equations
• Bispectrality and Time-Band-Limiting: Matrix valued polynomials
• The cone-beam transform and spherical convolution operators
• Time and band limiting for matrix valued functions: an integral and a commuting differential operator