Tauberian conditions, under which statistical convergence follows from statistical summability (C, 1) ✩

J.A. Fridly and M.K. Khan have recently extended Hardy’s and Landau’s Tauberian theorems to the case of statistical convergence, which was introduced by H. Fast in 1951. Let (xk: k = 0, 1, 2, . . .) be a sequence of real or complex numbers and set σn := (n + 1)−1 n k=0 xk for n = 0, 1, 2, . . . . We present necessary and sufficient conditions, under which st-limxk = L follows from st-limσn = L, where L is a finite number. If (xk) is a sequence of real numbers, then these are one-sided Tauberian conditions. If (xk) is a sequence of complex numbers, then these are two-sided Tauberian conditions. In particular, our conditions are satisfied if (xk) is statistically slowly decreasing (or increasing) in the case of real sequences; or if (xk) is statistically slowly oscillating in the case of complex sequences. Even these special sufficient conditions imply those given by Fridy and Khan.  2002 Elsevier Science (USA). All rights reserved.