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## Regular statistical convergence of double sequences

### Volume 102 / 2005

Colloquium Mathematicum 102 (2005), 217-227 MSC: Primary 40A05; Secondary 42B05. DOI: 10.4064/cm102-2-4

#### 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 $\{ x_{jk}: (j, k) \in {{\mathbb N}}^2\}$ is said to be regularly statistically convergent if (i) the double sequence $\{ x_{jk}\}$ is statistically convergent to some $\xi \in {{\mathbb C}}$, (ii) the single sequence $\{ x_{jk} : k\in {{\mathbb N}}\}$ is statistically convergent to some $\xi _j \in {{\mathbb C}}$ for each fixed $j\in {{\mathbb N}}\setminus {\mathcal S}_1$, (iii) the single sequence $\{ x_{jk} : j\in {{\mathbb N}}\}$ is statistically convergent to some $\eta _k\in {{\mathbb C}}$ for each fixed $k\in {{\mathbb N}}\setminus {\mathcal S}_2$, where ${\mathcal S}_1$ and ${\mathcal S}_2$ are subsets of ${{\mathbb N}}$ whose natural density is zero. We prove that under conditions (i)–(iii), both $\{ \xi _j\}$ and $\{ \eta _k\}$ are statistically convergent to $\xi$. As an application, we prove that if $f\in L \mathop {\rm log}\nolimits ^+ L({{\mathbb T}}^2)$, then the rectangular partial sums of its double Fourier series are regularly statistically convergent to $f(u,v)$ at almost every point $(u,v) \in {{\mathbb T}}^2$. Furthermore, if $f\in C({{\mathbb T}}^2)$, then the regular statistical convergence of the rectangular partial sums of its double Fourier series holds uniformly on ${{\mathbb T}}^2$.

#### Authors

• Ferenc MóriczBolyai Institute
University of Szeged