Some Applications of Summability Theory

Braha, Naim L.

doi:10.1007/978-981-10-0913-6_8

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“…The theory of Tauberian theorems was investigated intensively from many authors(see [1,2,3], [6,7], [9], [4]). In this paper our aim is to find conditions (so-called Tauberian) under which the converse implication holds, in Theorem 1.3, for defined convergence.…”

Section: Introductionmentioning

confidence: 99%

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

Braha¹,

Temaj²

2018

bspm

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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.

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