2014/11/24 by Jorge L. deLyra, deLyra, Jorge L.
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Matrix Theory and Algorithms #math-ph #math.CV #math.MP
paper · pdf · doi:10.48550/arxiv.1411.6503
44 pages, including 18 figures and 17 pages of appendices with explicit calculations and examples; made a few improvements in the text
openalex publication_date 2014/11/24 · arxiv created 2015/03/24 · arxiv updated 2015/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∞ 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.