Tauberian conditions under which statistical convergence follows from statistical summability $(EC)_{n}^1$

Abstract

Let $(x_k)$, for $k\in \mathbb{N}\cup \{0\}$ be a sequence of real or complex numbers and set $(EC)_{n}^{1}=\frac{1}{2^n}\sum_{j=0}^{n}{\binom{n}{j}\frac{1}{j+1}\sum_{v=0}^{j}{x_v}},$ $n\in \mathbb{N}\cup \{0\}.$ We present necessary and sufficient conditions, under which $st-\lim_{}{x_k}= L$ follows from $st-\lim_{}{(EC)_{n}^{1}} = L,$ where L is a finite number. If $(x_k)$ is a sequence of real numbers, then these are one-sided Tauberian conditions. If $(x_k)$ is a sequence of complex numbers, then these are two-sided Tauberian conditions.

A Tauberian theorem for the statistical generalized Nörlund-Euler summability method

“…The theory of Tauberian is extensively studied by many authors ( [1], [2], [3], [4], [7], [9]). In this section our aim is to find conditions (so-called Tauberian) under which the converse implication holds, for defined convergence.…”

“…Any theorem which states that convergence of a sequence follows from its N n p,q E 1 n summability and some Tauberian condition is said to be a Tauberian theorem for the N n p,q E 1 n summability method. The inclusion and Tauberian type theorems are proved in the papers [6,7,2,3,4,5,8], and some theorems of inclusion, Tauberian and convexity type for certain families of generalized Nörlund methods are obtained in [9].…”