Pythagorean Harmonic Summability of Fourier Series

TL;DR

This paper introduces a novel nonlinear summation method for Fourier series using Pythagorean harmonic mean, incorporating new concepts like smoothing operators and semi-harmonic summation, with applications to seismic signals.

Contribution

It presents new theoretical results and concepts such as the smoothing operator and semi-harmonic summation, expanding the framework of Fourier series summability.

Findings

Smoothing operator acts as Kalman filter for linear summability

Pythagorean harmonic summability is demonstrated with logistic processing

Regularization resolves inapplicability to seismic-like signals

Abstract

The paper explores the possibility for summing Fourier series nonlinearly via the Pythagorean harmonic mean. It reports on new results for this summability with introduction of new concepts like the smoothing operator and semi-harmonic summation. The smoothing operator is demonstrated to be Kalman filtering for linear summability, logistic processing for Pythagorean harmonic summability and linearized logistic processing for semi-harmonic summability. An emerging direct inapplicability of harmonic summability to seismic-like signals is shown to be resolvable by means of a regularizational asymptotic approach.