  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.
