Regular statistical convergence of double sequences
Volume 102 / 2005
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 ${{\mathbb T}}^2$.
