Low-Pass Filters, Fourier Series and Partial Differential Equations

TL;DR

This paper demonstrates that linear low-pass filters can regularize divergent Fourier series in physics applications involving PDEs, clarifying that divergences stem from oversimplified problem formulations rather than fundamental issues.

Contribution

It introduces a precise mathematical definition of linear low-pass filters, explores their properties, and shows their commutation with second derivatives, simplifying their use in physics.

Findings

Low-pass filters regularize divergent Fourier series.

Divergences are due to oversimplification, not fundamental physics.

First-order filter commutes with second derivatives.

Abstract

When Fourier series are used for applications in physics, involving partial differential equations, sometimes the process of resolution results in divergent series for some quantities. In this paper we argue that the use of linear low-pass filters is a valid way to regularize such divergent series. In particular, we show that these divergences are always the result of oversimplification in the proposition of the problems, and do not have any fundamental physical significance. We define the first-order linear low-pass filter in precise mathematical terms, establish some of its properties, and then use it to construct higher-order filters. We also show that the first-order linear low-pass filter, understood as a linear integral operator in the space of real functions, commutes with the second-derivative operator. This can greatly simplify the use of these filters in physics applications, and we give a few simple examples to illustrate this fact.