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2005 | 102 | 2 | 217-227
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Regular statistical convergence of double sequences

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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 ∖ 𝓢₁, (iii) the single sequence ${x_{jk}: j ∈ ℕ}$ is statistically convergent to some $η_k ∈ ℂ$ for each fixed $k ∈ ℕ ∖ 𝓢₂$, where 𝓢₁ and 𝓢₂ 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(𝕋²), then the rectangular partial sums of its double Fourier series are regularly statistically convergent to f(u,v) at almost every point (u,v) ∈ 𝕋². Furthermore, if f ∈ C(𝕋²), then the regular statistical convergence of the rectangular partial sums of its double Fourier series holds uniformly on 𝕋².
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  • Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm102-2-4
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