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When Fourier series are employed to solve partial differential equations,low-pass filters can be used to regularize divergent series that may appear. Inthis paper we show that the linear low-pass filters defined in a previous papercan 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 linearlow-pass filter corresponds to an operation in the complex plane that we referto as "singularity splitting", in which any given singularity of an inneranalytic function on the unit circle is replaced by two softer singularities onthat same circle, thus leading to corresponding DP Fourier series with betterconvergence characteristics. Higher-order linear low-pass filters can be easilydefined within the unit disk of the complex plane, in terms of the first-orderone. The construction of infinite-order filters, which always result in$C^{\infty}$ real functions over the unit circle, and in corresponding DPFourier series which are absolutely and uniformly convergent to thesefunctions, is presented and discussed.

Fourier Theory on the Complex Plane III: Low-Pass Filters, Singularity Splitting and Infinite-Order Filters