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

By F. Móricz. 2002

85 citations recorded by the source corpus.

Abstract 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 (x k : k=0,1,2,…) be a sequence of real or complex numbers and set σn:=(n+1)−1∑nk=0xk 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.

Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/927404e8695ff2eb41cf807276e4801dd0ab7b64