A theorem for the (J, p) summability method

Çanak, İbrahim; Totur, Ümi̇t

doi:10.1007/s10474-014-0452-y

Cited by 6 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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“…Tauberian theorems for continuous-type power series methods have a long history; see for example the books in [5,8] and related recent papers in [2,4,13,16]. The studies for the discrete-type summability methods began with the work of Armitage and Maddox [1] on discrete Abel means (A λ ) .…”

Section: Introductionmentioning

confidence: 99%