Introduction. This note investigates an apparent generalization of unconditionally convergent series ^ x » in weakly complete Banach spaces. A series of elements with Xi in E is said to be unconditionally convergent if for every variation of sign €j= ± 1 , ^TMeiXi is convergent. This formulation of the definition of unconditional convergence is equivalent to that given by Orliczjé]. We call ^Xi...

We improve and generalize a result on a local property of |T |k summability of factored Fourier series due to Sarıgöl [6].

In this paper, a general theorem concerning absolute matrix summability is established by applying the concepts of almost increasing and δ-quasi-monotone sequences.

Abstract The main object of this article is to introduce the concepts ( f, ρ )− statistical convergence order α and strong summability sequences real numbers give some inclusion relations between these spaces.

This article deals with some special cases which are extension of the strong summability of Fourier series with constant factor. We obtain a new equivalent form of inequalities A 2π 0 φ(e iθ) r dθ ≤ 2π 0 1 0 (1 − ρ) φ (z) 2 dρ r/2 dθ ≤ B 2π 0 φ(e iθ) r dθ, (1) 2π 0 1 0 (1 − ρ) q−1 φ (z) q dρ r/q dθ ≤ C 2π 0 φ(e iθ) r dθ, (2) D 2π 0 φ(e iθ) r dθ ≤ 2π 0 1 0 (1 − ρ) p−1 φ (z) p dρ r/p dθ.

Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitable sum rules defining C functions of the perturbation strength: here we find sufficient conditions for the Borel summability of their sums in t...