Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability

1995 ◽  
Vol 66 (1-2) ◽  
pp. 105-111 ◽  
Author(s):  
F. Móricz ◽  
B. E. Rhoades
Keyword(s):  
Weighted Mean ◽  
Tauberian Conditions ◽  
Necessary And Sufficient

Some Tauberian conditions obtained through weighted generator sequences

2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Cemal Belen
Keyword(s):  
Statistical Convergence ◽  
Weighted Mean ◽  
Tauberian Conditions ◽  
Weighted Means ◽  
Necessary And Sufficient

AbstractRecently, the concept of weighted generator sequence has been introduced by Çanak and Totur [Comput. Math. Appl. 62 (2011), no. 6, 2609–2615]. They proved that certain conditions in terms of weighted generator sequences are Tauberian conditions for the weighted mean method. In this paper, we present the necessary and sufficient Tauberian conditions based on a weighted generator sequence under which statistical convergence follows from statistical summability by weighted means.

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

2020 ◽  
Vol 24 (1) ◽  
pp. 49-57
Author(s):  
İbrahim Çanak ◽  
Gizem Erikli ◽  
Sefa Anıl Sezer ◽  
Ece Yaraşgil
Keyword(s):  
Tauberian Theorems ◽  
Normed Spaces ◽  
Weighted Mean ◽  
Tauberian Conditions ◽  
Necessary And Sufficient

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.

scholarly journals Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim's sense

2004 ◽  
Vol 2004 (65) ◽  
pp. 3499-3511 ◽  
Author(s):  
Ferenc Móricz ◽  
U. Stadtmüller
Keyword(s):  
Double Sequence ◽  
Complex Numbers ◽  
Double Sequences ◽  
Real Numbers ◽  
Weighted Mean ◽  
Tauberian Conditions ◽  
Slow Decrease ◽  
Necessary And Sufficient ◽  
Convergence In Pringsheim’S Sense

After a brief summary of Tauberian conditions for ordinary sequences of numbers, we consider summability of double sequences of real or complex numbers by weighted mean methods which are not necessarily products of related weighted mean methods in one variable. Our goal is to obtain Tauberian conditions under which convergence of a double sequence follows from its summability, where convergence is understood in Pringsheim's sense. In the case of double sequences of real numbers, we present necessary and sufficient Tauberian conditions, which are so-called one-sided conditions. Corollaries allow these Tauberian conditions to be replaced by Schmidt-type slow decrease conditions. For double sequences of complex numbers, we present necessary and sufficient so-called two-sided Tauberian conditions. In particular, these conditions are satisfied if the summable double sequence is slowly oscillating.

scholarly journals Necessary and sufficient conditions under which convergence follows from summability by weighted means

2001 ◽  
Vol 27 (7) ◽  
pp. 399-406 ◽  
Author(s):  
Ferenc Móricz ◽  
Ulrich Stadtmüller
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Real Numbers ◽  
Weighted Mean ◽  
Tauberian Conditions ◽  
Weighted Means ◽  
Necessary And Sufficient

We prove necessary and sufficient Tauberian conditions for sequences summable by weighted mean methods. The main results of this paper apply to all weighted mean methods and unify the results known in the literature for particular methods. Among others, the conditions in our theorems are easy consequences of the slowly decreasing condition for real numbers, or slowly oscillating condition for complex numbers. Therefore, practically all classical (one-sided as well as two-sided) Tauberian conditions for weighted mean methods are corollaries of our two main theorems.

scholarly journals Gap Tauberian theorems

1993 ◽  
Vol 47 (3) ◽  
pp. 385-393 ◽  
Author(s):  
Jeff Connor
Keyword(s):  
Tauberian Theorem ◽  
Strong Summability ◽  
Tauberian Theorems ◽  
Weighted Mean ◽  
Tauberian Condition ◽  
Support Set ◽  
Tauberian Conditions ◽  
Summability Matrix ◽  
Regular Summability ◽  
Bk Spaces

In the first section we establish a connection between gap Tauberian conditions and isomorphic copies of Co for perfect coregular conservative BK spaces and in the second we give a characterisation of gap Tauberian conditions for strong summability with respect to a nonnnegative regular summability matrix. These results are used to show that a gap Tauberian condition for strong weighted mean summability is also a gap Tauberian condition for ordinary weighted mean summability. We also make a remark regarding the support set of a matrix and give a Tauberian theorem for a class of conull spaces.

scholarly journals Abel-type weighted means transformations intoℓ

2000 ◽  
Vol 23 (3) ◽  
pp. 197-204
Author(s):  
Mulatu Lemma ◽  
George Tessema
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Weighted Mean ◽  
Weighted Means ◽  
Necessary And Sufficient

Letqk=(k+α    k)forα>−1andQn=∑k=0nqk. SupposeAq={ank}, whereank=qk/Qnfor0≤k≤nand 0 otherwise.Aqis called the Abel-type weighted mean matrix. The purpose of this paper is to study these transformations as mappings intoℓ. A necessary and sufficient condition forAqto beℓ-ℓis proved. Also some other properties of theAqmatrix are investigated.

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