scholarly journals Tauberian Theorems for the Product of Weighted and Cesàro Summability Methods for Double Sequences

2019 ◽  
Vol 38 (7) ◽  
pp. 9-19
Author(s):  
Gökşen Fındık ◽  
İbrahim Çanak
Keyword(s):  
Double Sequence ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Cesàro Summability ◽  
Tauberian Conditions ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Necessary And Sufficient

In this paper, we obtain necessary and sufficient conditions, under which convergence of a double sequence in Pringsheim's sense follows from its weighted-Cesaro summability. These Tauberian conditions are one-sided or two-sided if it is a sequence of real or complex numbers, respectively.

scholarly journals Convergence follows from Cesàro summability in the case of slowly decreasing or slowly oscillating double sequences in certain senses

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4489-4511
Author(s):  
Zerrin Önder ◽  
İbrahim Çanak
Keyword(s):  
Double Sequence ◽  
Complex Numbers ◽  
Tauberian Theorems ◽  
Finite Limit ◽  
Double Sequences ◽  
Cesàro Summability ◽  
Tauberian Conditions ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Special Cases

Let (u??) be a double sequence of real or complex numbers which is (C; 1; 1) summable to a finite limit. We obtain some Tauberian conditions of slow decreasing or oscillating types in terms of the generator sequences in certain senses under which P-convergence of a double sequence (u??) follows from its (C,1,1) summability. We give Tauberian theorems in which Tauberian conditions are of Hardy and Landau types as special cases of our results. We present some Tauberian conditions in terms of the de la Vall?e Poussin means of double sequences under which P-convergence of a double sequence (u??) follows from its (C,1,1) summability. Moreover, we give analogous results for (C,1,0) and (C,0,1) summability methods.

Summability methods which include the Riesz typical means. I

1971 ◽  
Vol 69 (1) ◽  
pp. 99-106 ◽  
Author(s):  
Dennis C. Russell
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternative Form ◽  
Cesàro Summability ◽  
Abel Summability ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Riesz Summability ◽  
Necessary And Sufficient

A number of special results exist for summability methods B which, include Riesz summability (R,λ,k)—for example, when B is generalized Abel summability (A,λ,ρ) [Kuttner(5)], or Riemann summability (,λ,μ) [Russell(14)], or Riemann-Cesàro summability (,λ,p,α) [Rangachari(12)], or generalized Cesàro summability (C,λ,k) [Meir (9); Borwein and Russell (l)]. The question of necessary and sufficient conditions to be satisfied by an arbitrary method B in order that B ⊇ (R,λ,k) has received an answer only for limited values of λ and k—for example, by Lorentz [(6), Theorem 10] for k = 1; the restrictions on λ in this case were removed by Maddox [(8), Theorem 1]. Thus (apart from the well-known case k = 0) the case k = 1 is the only one for which a complete solution exists, though application of a theorem of Russell [(13), Theorem 1A] yields one form of a result for 0 < k ≤ 1. Maddox's results, however, suggest an alternative form capable of generalization to all k ≥ 0, and in this paper we obtain a complete solution for 0 < k ≤ 1 in that form, without restriction on λ. We first recall the following definitions.

Tauberian conditions under which statistical convergence follows from statistical summability by weighted means

2004 ◽  
Vol 41 (4) ◽  
pp. 391-403 ◽  
Author(s):  
Ferenc Móricz ◽  
Cihan Orhan
Keyword(s):  
Finite Number ◽  
Large Class ◽  
Statistical Convergence ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Real Numbers ◽  
Tauberian Conditions ◽  
Summability Methods ◽  
Weighted Means ◽  
Necessary And Sufficient

The first named author has recently proved necessary and sufficient Tauberian conditions under which statistical convergence (introduced by H. Fast in 1951) follows from statistical summability (C, 1). The aim of the present paper is to generalize these results to a large class of summability methods (,p) by weighted means. Let p = (pk : k = 0,1, 2,...) be a sequence of nonnegative numbers such that po > 0 and Let (xk) be a sequence of real or complex numbers and set for n = 0,1, 2,.... We present necessary and sufficient conditions under which the existence of the limit st-lim xk = L follows from that of st-lim tn = L, where L is a finite number. If (xk) is a sequence of real numbers, then these are one-sided Tauberian conditions. If (xk) is a sequence of complex numbers, then these are two-sided Tauberian conditions.

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.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

Summability methods which include the Riesz typical means. II

1971 ◽  
Vol 69 (2) ◽  
pp. 297-300 ◽  
Author(s):  
B. C. Russell
Keyword(s):  
Regular Sequence ◽  
Cesàro Summability ◽  
Convergence Factor ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Factor Theorem ◽  
T Matrix ◽  
Image Position ◽  
Summability Matrix ◽  
Necessary And Sufficient

By making use of a convergence-factor theorem of Bosanquet(3), Cooke((4), Theorem I) gave conditions for a regular sequence-to-sequence summability matrix B to be at least as strong as Cesàro summability (C, κ) (κ > 0), namely:Theorem C. Let κ > 0. In order that the T-matrix B = (bρμ) shall satisfy B ⊇ (C, κ) it is necessary and sufficient thatIf 0 < κ ≤ 1 then (2) alone is necessary and sufficient.

scholarly journals Tauberian conditions under which convergence follows from Cesàro summability of double integrals over R2

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3425-3440
Author(s):  
Gökşen Fındık ◽  
İbrahim Çanak
Keyword(s):  
Continuous Function ◽  
The Other ◽  
Cesàro Summability ◽  
Double Integral ◽  
Tauberian Conditions ◽  
Summability Methods ◽  
Cesaro Summability ◽  
The Difference ◽  
Complex Valued ◽  
Double Integrals

For a real- or complex-valued continuous function f over R2+:= [0,1) x [0,1), we denote its integral over [0,u] x [0,v] by s(u,v) and its (C,1, 1) mean, the average of s(u,v) over [0,u] x [0,v], by ?(u,v). The other means (C,1,0) and (C; 0; 1) are defined analogously. We introduce the concepts of backward differences and the Kronecker identities in different senses for double integrals over R2+. We give onesided and two-sided Tauberian conditions based on the difference between double integral of s(u, v) and its means in different senses for Ces?ro summability methods of double integrals over [0,u] x [0,v] under which convergence of s(u,v) follows from integrability of s(u,v) in different senses.

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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