On the commutation properties of finite convolution and differential operators I: commutation

TL;DR

This paper characterizes all finite convolution operators that commute with differential operators, extending previous results to non-symmetric kernels and exploring implications for spectral properties.

Contribution

It provides a comprehensive characterization of convolution operators commuting with differential operators without symmetry constraints, broadening prior understanding.

Findings

Identifies all convolution operators commuting with differential operators.

Extends Morrison's results to non-symmetric kernels.

Implications for spectral analysis of convolution operators.

Abstract

The commutation relation $KL = LK$ between finite convolution integral operator $K$ and differential operator $L$ has implications for spectral properties of $K$. We characterize all operators $K$ admitting this commutation relation. Our analysis places no symmetry constraints on the kernel of $K$ extending the well-known results of Morrison for real self-adjoint finite convolution integral operators.