A Tauberian Theorem for the (C, 1)(N, 1/(n + 1)) Summability Method

1970 ◽  
Vol 25 (2) ◽  
pp. 391 ◽  
Author(s):  
H. P. Dikshit
Keyword(s):  
Tauberian Theorem ◽  
Summability Method

Classical Tauberian Theorems for the (C; 1; 1) Summability Method

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Subject Classification ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Littlewood Theorem

Abstract In this paper we generalize some classical Tauberian theorems for single sequences to double sequences. One-sided Tauberian theorem and generalized Littlewood theorem for (C; 1; 1) summability method are given as corollaries of the main results. Mathematics Subject Classification 2010: 40E05, 40G0

A Tauberian Theorem for Borel-Type Methods of Summability

1969 ◽  
Vol 21 ◽  
pp. 740-747 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Tauberian Theorem ◽  
Convergent Series ◽  
Summability Method ◽  
Negative Integer ◽  
Exponential Method ◽  
Image Position ◽  
Borel Type ◽  
Cesàro Method

Suppose throughout that α >0, β is real, and Nis a non-negative integer such that αN+ β> 0. A series of complex terms is said to be summable (B, α,β) to l if, as x→ ∞,where sn= a0 + a1 + … + an.The Borel-type summability method (B, α, β) is regular, i.e., all convergent series are summable (B, α,β) to their natural sums; and (B,1, 1) is the standard Borel exponential method B.Our aim in this paper is to prove the following Tauberian theorem.THEOREM. Iƒ(i) p ≧ – ½, an = o(np), and(ii) is summable (B, α,β) to l, then the series is summable by the Cesaro method(C, 2p + 1) to l.

scholarly journals A Tauberian Theorem for a General Summability Method

2015 ◽  
Vol 61 (1) ◽  
pp. 123-128
Author(s):  
Ibrahim Çanak
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Abel Summability ◽  
General Summability

Abstract We investigate conditions under which Mϕ summability implies Abel summability and give the generalized Littlewood Tauberian theorem for Mϕ summability method.

A Tauberian Theorem for the General Euler-Borel Summability Method

1992 ◽  
Vol 44 (5) ◽  
pp. 1100-1120 ◽  
Author(s):  
Laying Tam
Keyword(s):  
Positive Integer ◽  
Unit Disk ◽  
Tauberian Theorem ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Borel Summability ◽  
Closed Unit Disk ◽  
Tauberian Condition ◽  
Image Position ◽  
General Conditions

AbstractOur main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m > n, (m— n)n-p(f)→0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.

scholarly journals Tauberian conditions for Conull spaces

1985 ◽  
Vol 8 (4) ◽  
pp. 689-692 ◽  
Author(s):  
J. Connor ◽  
A. K. Snyder
Keyword(s):  
Tauberian Theorem ◽  
Convergent Sequence ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Tauberian Conditions

The typical Tauberian theorem asserts that a particular summability method cannot map any divergent member of a given set of sequences into a convergent sequence. These sets of sequences are typically defined by an “order growth” or “gap” condition. We establish that any conull space contains a bounded divergent member of such a set; hence, such sets fail to generate Tauberian theorems for conull spaces.

scholarly journals A Tauberian theorem for the discrete Mφ summability method

2012 ◽  
Vol 25 (4) ◽  
pp. 771-774
Author(s):  
İbrahim Çanak ◽  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Summability Method

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

2019 ◽  
Vol 11 (2) ◽  
pp. 251-263
Author(s):  
Naim L. Braha
Keyword(s):  
Tauberian Theorem ◽  
Sufficient Conditions ◽  
Summability Method ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Necessary And Sufficient

Abstract Let (pn) and (qn) be any two non-negative real sequences with {{\rm{R}}_{\rm{n}}}: = \sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n}} - {\rm{k}}}}} \ne 0\,\,\,\,\left( {{\rm{n}} \in {\rm\mathbb{N}}} \right) With {\rm{E}}_{\rm{n}}^1 − we will denote the Euler summability method. Let (xn) be a sequence of real or complex numbers and set {\rm{N}}_{{\rm{p}},{\rm{q}}}^{\rm{n}}{\rm{E}}_{\rm{n}}^1: = {1 \over {{{\rm{R}}_{\rm{n}}}}}\sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n - k}}}}{1 \over {{2^{\rm{k}}}}}\sum\limits_{{\rm{v}} = 0}^{\rm{k}} {\left( {_{\rm{v}}^{\rm{k}}} \right){{\rm{x}}_{\rm{v}}}} } for n ∈ ℕ. In this paper, we present necessary and sufficient conditions under which the existence of the st− limit of (xn) follows from that of {\rm{st - N}}_{{\rm{p}},q}^{\rm{n}}{\rm{E}}_{\rm{n}}^1 − limit of (xn). These conditions are one-sided or two-sided if (xn) is a sequence of real or complex numbers, respectively.

scholarly journals A Tauberian Theorem for Double Cesàro Summability Method

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
Bidu Bhusan Jena ◽  
Susanta Kumar Paikray ◽  
Umakanta Misra
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
De La Vallée Poussin

We have generalized Littlewood Tauberian theorems for(C,k,r)summability of double sequences by using oscillating behavior and de la Vallée-Poussin mean. Further, the generalization of(C,r)summability from(C,k,r)summability is given as corollaries which were earlier established by the authors.

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