Abstract
Abstract 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.
Highlights
It is well known that most 2L-periodic real-valued f (x) ∈ C[0, 2L] can be represented by the Fourier series f (x) ≔ S(x) = a0 2 +∑k∞=1 ak cos k π L x bk sin (1)with the partial sums, see e.g. [1], Sn(x) = n∑k=1 ak cos k π L
During the period 1990–2017, contemporary summability research has ranged from factored Fourier series [11] and product summability [12] of these series to generalizations for any orthogonal series, via sums based on Marcinkiewicz’s Θ-means [13,14]
Unique linear processing and contraction mapping features are identified for the smoothing operator of this summability
Summary
It is well known that most 2L-periodic real-valued f (x) ∈ C[0, 2L] can be represented by the Fourier series f (x). During the period 1990–2017, contemporary summability research has ranged from factored Fourier series [11] and product summability [12] of these series to generalizations for any orthogonal series, via sums based on Marcinkiewicz’s Θ-means [13,14]. Unique linear processing and contraction mapping features are identified for the smoothing operator of this summability
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