A Spectral Theory of Scalar Volterra Equations

Speaker: George Stepaniants (Caltech)

Date: 3/10/25

Abstract: Volterra integral equations have been the subject of much study, from the perspective of both pure mathematics and applied science. Developments in analysis have yielded far-ranging existence, uniqueness, and spectral results for such equations. In applied science, Volterra equations are natural models for signal convolution or filtering, and they describe the evolution of a variety of partially-observed, time-dependent systems. In particular, the study of strain-stress dynamics in materials science has inspired closed-form solutions to special classes of Volterra equations with exponentially decaying memory, as well as certain Volterra equations involving fractional derivatives and Prony series. Only a limited number of these results have been proven rigorously, however, and their study in materials science has remained largely disjoint from the broader mathematical community. In this talk, we develop a spectral theory for scalar, linear Volterra equations, showing that a variety of disparate results in applied science (including the aforementioned results in viscoelasticity) are special cases of a more general theory. In particular, we derive analytic solutions for large classes of continuous or discrete-time linear Volterra equations, as well as fractional and delay differential equations. We show how our closed-form solutions can be realized numerically with rational approximation, study the analytical properties of these formulas, and test them on a wide array of problems relating to signal deconvolution, triangular matrix inversion, and interconversion of materials.