Applications of the deferred de la Vallée Poussin means of Fourier series

In this paper, we have proved four theorems on deferred generalized De la Vallée Poussin summability of Fourier series. Some results obtained previously by others, pertaining to generalized De la Vallée Poussin summability of Fourier series, are particular cases of ours.

Pointwise (H, Φ) Strong Approximation by Fourier Series of LΨ Integrable Functions

We essentially extend and improve the classical result of G. H. Hardy and J. E. Littlewood on strong summability of Fourier series. We will present an estimation of the generalized strong mean (H, Φ) as an approximation version of the Totik type generaliza- tion of the result of G. H. Hardy, J. E. Littlewood, in case of integrable functions from LΨ. As a measure of such approximation we will use the function constructed by function Ψ com- plementary to Φ on the base of definition of the LΨ points. Some corollary and remarks will also be given.

Approximation of Signals by Harmonic-Euler Triple Product Means

Our paper deals with the approximation of signals by <em>H₁.E^θ.E^θ</em> product means of Fourier and its conjugate series. New theorems based on <em>H₁.E^θ.E^θ</em> product summability have been established and proved under general conditions. The established theorems extend, generalize and improve various existing results on summability of Fourier series and its conjugate series.

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.

Strong Riesz summability of Fourier series

The notion of strong summability was introduced by Fekete (Math. És Termesz Ertesitö, 34 (1916), 759-786). Dealing with Nörlund summability of Fourier series Mittal (J. Math. Anal. Appl. 314 (2006), 75-84) has established a result on strong summability. We have established a new result on sufficient condition for strong Riesz summability of Fourier series.