Low-Pass Filters, Fourier Series and
Partial Differential Equations
Jorge L. deLyra
Department of Mathematical Physics,
Physics Institute,
University of São Paulo
March 4, 2015
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.