On the weighted generator of double sequences and its Tauberian conditions

In this paper, our aim is tomake a novel interpretation of the relation between ((N) over bar, p, q) method and P-convergence for double sequences. In accordance with this aim, we derive some Tauberian conditions, controlling O-L- and O-oscillatory behavior of the weighted generator sequence of (u(mn)) in the sense (1, 1) relative to (P-m) and/or (Q(n)), from ((N) over bar, p, q) summability to P-convergence with some restrictions on the weight sequences. As special cases, we indicate that O-L-condition of Landau type with respect to (P-m) and (Q(n)) and O-condition of Hardy type with respect to (P-m) and (Q(n)) are Tauberian conditions for ((N) over bar, p, q) summability under some additional conditions. Therefore, these results contain all of the classical Tauberian theorems including slow decrease and slow oscillation conditions in certain senses.