A Spectral Theory of Scalar Volterra Equations (2503.06957v2)

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 (viscoelastic models); equations with positive definite kernels (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. It also introduces a geometric construction of the regularized Hilbert transform, revealing a fundamental connection to fractional and delay differential equations. Finally, it offers a practical toolbox for Volterra equations of all classes, reducing many to pen-and-paper calculations and introducing a powerful spectral approach for general Volterra equations based on rational approximation.