Some Tauberian Theorems for the Weighted Mean Method of Summability of Double Sequences

Let p = (p_j) and q = (q_k) be real sequences of nonnegative numbers with the property that

\(\begin{array}{ccccccc}{P}_{m}=\sum_{j=0}^{m}{p}_{j}\ne 0& {\text{and}}& {Q}_{m}=\sum_{k=0}^{n}{q}_{k}\ne 0& \mathrm{for all}& m& {\text{and}}& n.\end{array}\) $\begin{array}{ccccccc}{P}_m={\sum }_{j=0}^m{p}_j\ne 0& \text{and}& Q_m={\sum }_{k=0}^n{q}_k\ne 0& \mathrm{for}\mathrm{all}& m& \text{and}& n.\end{array}$

Also let (P_m) and (Q_n) be regularly varying positive indices. Assume that (u_mn) is a double sequence of complex (real) numbers, which is ( \(\overline{N }\) $\overline{N}$ , p, q; α, β)-summable and has a finite limit, where (α, β) = (1, 1), (1, 0), or (0, 1). We present some conditions imposed on the weights under which (u_mn) converges in Pringsheim’s sense. These results generalize and extend the results obtained by the authors in [Comput. Math. Appl., 62, No. 6, 2609–2615 (2011)].