TAUBERIAN THEOREMS IN THE STATISTICAL SENSE FOR THE WEIGHTED MEANS OF DOUBLE SEQUENCES

Abstract

Let $p := \{p_j\}_{j=0}^{\infty}$ and $q := \{q_k\}_{k=0}^{\infty}$ be complex sequences with $p,q \in$ SVA. Assume that $\{s_{mn}\}_{m,n=0}^\infty$ is a double sequence in $\mathbf{C}$ (or one of $\mathbf{R}$, a Banach space, and an ordered linear space), with $s_{mn} \stackrel{st}{\rightarrow} s \hspace{6 pt} (\bar{\mathrm{N}},p,q;\alpha,\beta)$, where $(\alpha,\beta) = (1,1)$, $(1,0)$ or $(0,1)$. We give sufficient and/or necessary conditions under which $s_{mn} \stackrel{st}{\rightarrow} s$. The theory developed here is the statistical version of the work of Chen and Hsu in [Anal. Math. 26 (2000), 243-262]. Our results generalize Móricz, [J. Math. Anal. Appl. 286 (2003), 340-350].