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

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