  When Fourier series are employed to solve partial differential equations,
low-pass filters can be used to regularize divergent series that may appear. In
this paper we show that the linear low-pass filters defined in a previous paper
can be interpreted in terms of the correspondence between Fourier Conjugate
(FC) pairs of Definite Parity (DP) Fourier series and inner analytic functions,
which was established in earlier papers. The action of the first-order linear
low-pass filter corresponds to an operation in the complex plane that we refer
to as "singularity splitting", in which any given singularity of an inner
analytic function on the unit circle is replaced by two softer singularities on
that same circle, thus leading to corresponding DP Fourier series with better
convergence characteristics. Higher-order linear low-pass filters can be easily
defined within the unit disk of the complex plane, in terms of the first-order
one. The construction of infinite-order filters, which always result in
$C^{\infty}$ real functions over the unit circle, and in corresponding DP
Fourier series which are absolutely and uniformly convergent to these
functions, is presented and discussed.
