A Spectral Theory of Scalar Volterra Equations

Abstract

This work aims to bridge the gap between pure and applied research on scalar, linear Volterra equations by examining five major classes: integral and integro-differential equations with completely monotone kernels, such as linear viscoelastic models; equations with positive definite kernels, such as partially observed quantum systems; difference equations with discrete, positive definite kernels; a generalized class of delay differential equations; and a generalized class of fractional differential equations. We develop a general, spectral theory that provides a system of correspondences between these disparate domains. As a result, we see how 'interconversion' (operator inversion) arises as a natural, continuous involution within each class, yielding a plethora of novel formulas for analytical solutions of such equations. This spectral theory unifies and extends existing results in viscoelasticity, signal processing, and analysis, and makes progress on an open question of Abel regarding the solution of integral equations of the first kind. Finally, it reduces simple Volterra equations of all classes to pen-and-paper calculation, and offers promising applications to the numerical solution of Volterra equations more broadly.