Tauberian theorems for double sequences that are statistically summable (C, 1, 1) ✩

Let (xjk: j, k = 0, 1, 2, . . .) be a double sequence of real or complex numbers, and set σmn := (m +1)−1(n+1)−1 m j=0 n k=0 xjk for m,n = 0, 1, 2, . . . . We give necessary and sufficient conditions, under which st-limσmn = ξ implies st-lim xjk = ξ, where ξ is a finite number. These Tauberian conditions are one-sided when the xjk are real numbers, and they are two-sided when the xjk are complex numbers. In particular, these Tauberian conditions are clearly satisfied if (xjk) is statistically slowly decreasing in the case of real sequences or if (xjk) is statistically slowly oscillating in the case of complex sequences.  2003 Elsevier Inc. All rights reserved.