  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.
