Four dimensional matrix mappings and applications

In this paper, we characterize the classes (L,Lk) , (Lk,L) and (L∞,Lk) , 1 ≤ k < ∞, of all four dimensional infinite matrices, where Lk and L∞ are the spaces of all absolutely k-summable and bounded double sequences, respectively. Using them, we establish some relations between N, pn, qn and N, p′ n, q′n k summability methods which extend some results of Bosanquet (1950), Sarıg¨ol (1993), Sarıg¨ol and Bor (1995), and Sunouchi (1949) to double summability methods, and give a relation between single and double summability methods.