Pythagorean harmonic summability of Fourier series

This paper explores the possibility for summing Fourier series nonlinearly via the Pythagorean harmonic mean. It reports on new results for this summability with the 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.