Some general Tauberian conditions for the weighted mean summability method

Totur, Ümi̇t; Çanak, İbrahim

doi:10.1016/j.camwa.2011.12.005

Cited by 17 publications

(9 citation statements)

References 10 publications

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“…The inclusion and Tauberian type theorems are proved in the papers [7,8], and some inclusion, Tauberian and convexity type theorems for certain families of generalized Nörlund methods are obtained in [12]. Moreover, some theorems of Abelian and Tauberian type have been recently proved for the methods of power series, (C, α) and weighted mean in [2][3][4][5]13].…”

Section: If Limmentioning

confidence: 99%

A Tauberian theorem for the generalized Nörlund summability method

Çanak

Braha²,

Totur

2018

Georgian Mathematical Journal

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Let (p n ) and (q n ) be any two non-negative real sequences, with R n := ∑ n k=0 p k q n−k ̸ = 0 (n ∈ ℕ). Let ∑ ∞ k=0 a k be a series of real or complex numbers with partial sums (s n ), and set t p,q n := 1 R n ∑ n k=0 p k q n−k s k for n ∈ ℕ. In this paper, we present the necessary and sufficient conditions under which the existence of the limit lim n→∞ s n = L follows from that of lim n→∞ t p,q n = L. These conditions are one-sided or two-sided if (s n ) is a sequence of real or complex numbers, respectively. MSC 2010: 40G15, 41A36We define the following sets:A(n, t) := {q λ n −k − q n−k : k = 0, 1, 2, . . . , n, λ > 1}, B(n, t) := {q k−λ n − q n−k : k = 0, 1, 2, . . . , t n , 0 < λ < 1}, where λ n := [λn] denotes the integral part of λn.

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Section: If Limmentioning

confidence: 99%

A Tauberian theorem for the generalized Nörlund summability method

Çanak

Braha²,

Totur

2018

Georgian Mathematical Journal

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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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“…For each integer m ≥ 0, we define σ Totur and Ç anak [16,Lemma 1] proved the identity that for a sequence (u n ) and any integer m ≥ 1,…”

Section: Introductionmentioning

confidence: 99%

“…n,p (ω (m−1) (u)). The weighted general control modulo of order m of (u n ) has been used recently as Tauberian conditions for various summability methods in [1][2][3]16].…”

Section: Introductionmentioning

confidence: 99%