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

Jorge L. deLyra|arXiv (Cornell University)|Nov 24, 2014

Algebraic and Geometric Analysis2 references3 citations

TL;DR

This paper introduces a complex-plane framework for understanding low-pass filters in Fourier analysis, showing that first-order filters perform 'singularity splitting'—replacing a single singularity on the unit circle with two milder ones—thereby improving convergence of Definite Parity Fourier series. Higher-order and infinite-order filters are constructed via iteration, yielding $C^{rown ext{infty}}$ functions with absolutely and uniformly convergent Fourier series.

ABSTRACT

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.

Motivation & Objective

• To interpret linear low-pass filters in the context of inner analytic functions on the unit disk.
• To address convergence issues in divergent Definite Parity (DP) Fourier series arising in PDE solutions.
• To develop a geometric and analytic framework for higher-order and infinite-order filters via singularity manipulation.
• To demonstrate that infinite-order filters produce $C^{rown\text{infty}}$ real functions with uniformly convergent DP Fourier series.

Proposed method

• Utilizes the correspondence between Fourier Conjugate (FC) pairs of DP Fourier series and inner analytic functions in the open unit disk.
• Defines first-order low-pass filters via convolution with a rectangular kernel $K_{ heta}^{(1)}$, which has compact support and unit integral.
• Constructs higher-order filters by iteratively applying the first-order filter, leading to smoother kernels with increasing differentiability.
• Derives the infinite-order filter as the limit of higher-order filters, resulting in a $C^{rown\text{infty}}$ kernel with dense singularities on the unit circle.
• Analyzes the analyticity of the kernel function by examining Taylor series convergence around points with vanishing high-order derivatives.
• Demonstrates that the kernel is not analytic at any point in the support interval due to non-representability by convergent Taylor series.

Experimental results

Research questions

• RQ1 How can first-order low-pass filters be interpreted geometrically in the complex plane using inner analytic functions?
• RQ2 What is the mechanism by which singularity splitting improves the convergence of Definite Parity Fourier series?
• RQ3 How can higher-order filters be systematically constructed within the unit disk framework?
• RQ4 What are the analytic properties of the infinite-order filter kernel, and can it be extended analytically to the complex plane?
• RQ5 Why is the infinite-order kernel not analytic despite being $C^{rown\text{infty}}$?

Key findings

• The first-order low-pass filter corresponds to 'singularity splitting'—a single singularity on the unit circle is replaced by two softer singularities, enhancing series convergence.
• Higher-order filters are constructed by iterative application of the first-order filter, resulting in kernels with increasing smoothness and vanishing derivatives at more points.
• The infinite-order filter kernel is $C^{rown\text{infty}}$ and has a set of $2^n + 1$ points in its support where all derivatives of order $n$ or higher vanish, with this set becoming dense as $n \to \infty$.
• The infinite-order kernel is not analytic at any point in its support interval because it cannot be represented by a convergent Taylor series around any such point.
• The kernel function cannot be analytically continued to the complex plane via standard analytic continuation, due to a dense set of non-analytic points on the unit circle.
• The resulting DP Fourier series for the infinite-order filter are absolutely and uniformly convergent to $C^{rown\text{infty}}$ real functions on the unit circle.