Regular statistical convergence of double sequences

Ferenc Móricz. "Regular statistical convergence of double sequences." Colloquium Mathematicae 102.2 (2005): 217-227. <http://eudml.org/doc/283994>.

@article{FerencMóricz2005,
abstract = {The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence $\{x_\{jk\}: (j,k) ∈ ℕ²\}$ is said to be regularly statistically convergent if (i) the double sequence $\{x_\{jk\}\}$ is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence $\{x_\{jk\} : k ∈ ℕ\}$ is statistically convergent to some $ξ_j ∈ ℂ$ for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence $\{x_\{jk\}: j ∈ ℕ\}$ is statistically convergent to some $η_k ∈ ℂ$ for each fixed $k ∈ ℕ ∖ ₂$, where ₁ and ₂ are subsets of ℕ whose natural density is zero. We prove that under conditions (i)-(iii), both $\{ξ_j\}$ and $\{η_k\}$ are statistically convergent to ξ. As an application, we prove that if f ∈ L log⁺L(²), then the rectangular partial sums of its double Fourier series are regularly statistically convergent to f(u,v) at almost every point (u,v) ∈ ². Furthermore, if f ∈ C(²), then the regular statistical convergence of the rectangular partial sums of its double Fourier series holds uniformly on ².},
author = {Ferenc Móricz},
journal = {Colloquium Mathematicae},
keywords = {convergence in Pringsheim's sense; regular convergence; statistical convergence; natural density; statistical boundedness; regular statistical convergence; absolute convergence; double Fourier series; rectangular partial sum},
language = {eng},
number = {2},
pages = {217-227},
title = {Regular statistical convergence of double sequences},
url = {http://eudml.org/doc/283994},
volume = {102},
year = {2005},
}

TY - JOUR
AU - Ferenc Móricz
TI - Regular statistical convergence of double sequences
JO - Colloquium Mathematicae
PY - 2005
VL - 102
IS - 2
SP - 217
EP - 227
AB - The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence ${x_{jk}: (j,k) ∈ ℕ²}$ is said to be regularly statistically convergent if (i) the double sequence ${x_{jk}}$ is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence ${x_{jk} : k ∈ ℕ}$ is statistically convergent to some $ξ_j ∈ ℂ$ for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence ${x_{jk}: j ∈ ℕ}$ is statistically convergent to some $η_k ∈ ℂ$ for each fixed $k ∈ ℕ ∖ ₂$, where ₁ and ₂ are subsets of ℕ whose natural density is zero. We prove that under conditions (i)-(iii), both ${ξ_j}$ and ${η_k}$ are statistically convergent to ξ. As an application, we prove that if f ∈ L log⁺L(²), then the rectangular partial sums of its double Fourier series are regularly statistically convergent to f(u,v) at almost every point (u,v) ∈ ². Furthermore, if f ∈ C(²), then the regular statistical convergence of the rectangular partial sums of its double Fourier series holds uniformly on ².
LA - eng
KW - convergence in Pringsheim's sense; regular convergence; statistical convergence; natural density; statistical boundedness; regular statistical convergence; absolute convergence; double Fourier series; rectangular partial sum
UR - http://eudml.org/doc/283994
ER -