Necessary and sufficient Tauberian conditions for weighted mean methods of summability in two-normed spaces

Abstract

We first define the concept of weighted mean method of summability and then present necessary and sufficient Tauberian conditions for the weighted mean summability of sequences in two-normed spaces. As corollaries, we establish two-normed analogues of two classical Tauberian theorems.

Tauberian conditions under which statistical convergence follows from statistical summability by weighted mean method in two-normed spaces

“…Recently, Loku et al [15] established necessary and sufficient Tauberian conditions under which ordinary convergence of a sequence follows from its summability by Nörlund mean method in two-normed spaces. Although a special case of summability by Nörlund mean method is summability by weighted mean method, this present paper is roughly a statistical extension of the results in C ¸anak et al [2]. Therefore, the results obtained in Loku et al [15] and this present paper are completely independent of each other.…”

“…Theorem 1 is the statistical extension of the Tauberian theorem given in [2]. Condition (2.2) is equivalent to the following: For given ε > 0 and η > 0, there exists some λ > 1 such that lim sup for every 0 < λ < 1 and for every z ∈ X.…”

“…Savas ¸and Sezer [18] introduced the concept of Cesàro summability method and obtained necessary and sufficient Tauberian conditions for Cesàro summability method in two-normed spaces. Following Savas ¸and Sezer [18], C ¸anak et al [2] introduced the concept of weighted mean method of summability and established necessary and sufficient Tauberian conditions for the weighted mean summability of sequences in two-normed spaces.…”

“…In C ¸anak et al [2], Tauberian theorems for summability by weighted mean method were obtained in two-normed spaces. Recently, Loku et al [15] established necessary and sufficient Tauberian conditions under which ordinary convergence of a sequence follows from its summability by Nörlund mean method in two-normed spaces.…”

Tauberian theorems for Nörlund-Cesáro mean via statistically convergence in m-Normed spaces

“…Notice that (1.1) may imply (1.2) under a certain condition, which is called a Tauberian condition. For further results of Tauberian type theorems, a reader is referred to the following references: Braha [2-5,7], Canak, Braha and Totur [8], Canak, Erikli, Sezer and Yarasgil [9], Kiesel [16], Kiesel, Stadtmüller [17], Loku, Braha, Et, Tato [18], Loku, Braha [19]. Very recently, Savas, Sezer [21] and Braha, Loku [6], have studied the Tauberian Theorems in the 2-normed spaces.…”