On the Numerical Solutions to a Certain Sturm-Liouville ODE for Spectral Filtering Applications *

Spectral filtering has been recently introduced as a new approach to predicting a sequence of the outputs generated by dynamical systems. The spectral filtering comes with theoretical performance guarantees, quantifying how good the predictions are. The main advantage of spectral filtering is that the filters are fixed and can be computed completely offline before starting the prediction procedure. Indeed, the filters are the eigenvectors of a known Hankel matrix, which is ill-conditioned. In this paper, we prove that the eigenvectors can be approximated by solutions to a Sturm-Liouville ordinary differential equation. The Sturm-Liouville ODE contains an unknown parameter, which hugely impacts the approximated eigenvectors and their similarity with the eigenvectors obtained from the Hankel matrix. We discuss the properties of the Sturm-Liouville ordinary differential equation, and specify the boundary conditions and domain. In addition, we introduce an orthogonality measure which quantifies the quality of the estimated solutions and could be used to form initial guesses for the unknown parameter involved in the Sturm-Liouville ordinary differential equation. The code to solve the Sturm-Liouville ordinary equation is shared on a public GitHub page<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup>.