Regular statistical convergence of double sequences

Ferenc Móricz

Colloquium Mathematicae, Tome 103 (2005), p. 217-227 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2005-01-01

Zbl 1062.40002

EUDML-ID : urn:eudml:doc:283994