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Some Tauberian Theorems under Triple Statistically Nörlund-Cesáro Summability Method

Written By

Carlos Granados

Reviewed: 28 June 2022 Published: 04 August 2022

DOI: 10.5772/intechopen.106141

From the Edited Volume

Functional Calculus - Recent Advances and Development

Edited by Hammad Khalil

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In this paper, we extend the notion presented by Braha (2020) in a higher dimension, we introduce the notion of Np,qn,m,gCn,m,g1,1,1-statistically convergence and show necessity and sufficiency conditions under which the existence of the limit st-limn,m,g→∞xn,m,g=L follows from that st-limn,m,g→∞Np,qn,m,gCn,m,g1,1,1=L. These conditions are one-sided or two-sided if xn,m,g is a sequence of real or complex numbers, respectively.


  • Nörlund-Cesro summability method
  • one-sided and two-sided Tauberian conditions
  • triple statistical convergence

1. Introduction

The concept of statistical convergence was introduced by Fast [1] and Steinhaus [2]. Besides, in this connection, Fridy [3] showed some relation to a Tauberian condition for the statistical convergence of xk. Subsequently, many researchers have worked in this area in several settings. For more recent works in this direction, one may refer to [4, 5]. Existing works in this field based on statistical convergence appears to have been restricted to real or complex sequences; however, Parida et al. [6] extended the idea for a locally convex Hausdorff topological linear space. Tauber [7] introduced the first Tauberian theorems for single sequences, that an Abel summable sequence is convergent with some suitable conditions. Later, a huge number of authors such as Landau [8], Hardy and Littlewood [9], and Schmidt [10] obtained some classical Tauberian theorems for Cessáro and Abel summability methods of single sequences. Recently, Braha [11] introduced some notions on statistical convergence by using the Nörlund-Cesáro summability method in a single sequence and proved some Tauberian theorems. In the last year, Canak and Totur [12], and Jena et al. [13] presented and studied several Tauberian theorems for single sequences. On the other hand, Knopp [14] obtained some classical type Tauberian theorems for Abel and C,1,1 summability methods of double sequences and proved that Abel and C,1,1 summability methods hold for the set of bounded sequences. Further, Moricz [15] proved some Tauberian theorems for Cesáro summable double sequences and deduced Tauberian theorems of Landau [16] and Hardy [17] type. Canak and Totur [18] have proved a Tauberian theorem for Cesáro summability of single integrals and also the alternative proofs of some classical type Tauberian theorems for the Cesáro summability of single integrals and later introduced by Parida et al. [6] for double integrals. Otherwise, the notion of C,1,1,1 summability of a triple sequence was originally introduced by Canak and Totur in 2016 [19]. Later, Canak et al. [20] studied some C,1,1,1 means of a statistical convergent triple sequence and gave some classical Tauberian theorems for a triple sequence that P-convergence follows from statistically C,1,1,1 summability under the two-sided boundedness conditions and slowly oscillating conditions in certain senses. Then, in 2020 Totur and Canak [21] proved Tauberian conditions under which convergence of triple integrals follows from C,1,1,1 summability. For more studies associated to the main topic of this paper, we refer the reader to [22, 23, 24].

Let pn,m,g and qn,m,g be any two non-negative real sequences with


and C,1,1,1-Cesáro summability method. Let xn,m,g be a sequence of real of complex numbers and set


for nmg××.

In this paper, we show necessary and sufficient conditions under which the existence of the limit limn,m,gxn,m,g=L follows from that of limn,m,gNp,qn,m,gCn,m,g1,1,1=L. These conditions are one-sided or two-sided if xn,m,g is a sequence of real or complex numbers, respectively.

Given two non-negative sequences pn,m,g and qn,m,g, the convolution pq is defined by


with C,1,1,1 we will denote the triple Cesáro summability method. Now, let xn,m,g be a sequence, when pqn,m,g0 for all nmg×× the generalized Nörlund-Cesáro transform of the sequence xn,m,g is the sequence Np,qn,m,gCn,m,g1,1,1 obtained by putting


We say that the sequence xn,m,g is generalized Nörlund-Cesáro summable to L determined by the sequences pn,m,g and qn,m,g (or simply summable Np,qn,m,gCn,m,g1,1,1) to L if


Throughout this paper, we will assume that the sequences pn,m,g and qn,m,g are satisfying the following conditions


where λn=λn, λm=λm and λg=λg. On the other hand, an,m,gbn,m,g means that there are constants C,C1 such that an,m,gCbn,m,gC1an,m,g. If


implies (2), then the method Np,qn,m,gCn,m,g1,1,1 is said to be regular. Nevertheless, the converse is not always true as can be seen in the following example:

Let us consider that pn,m,g=qn,m,g=1 for all nmg××. Besides, we define the following sequence x=xi,j,k=1i+j+k, then we get


and as we know, x=xi,j,k is not convergent. Notice that (6) can imply (2) under a certain condition, which is called a Tauberian conditions. Any theorem which states that convergence of a sequence follows from its Np,qn,m,gCn,m,g1,1,1 summability and some Tauberian conditions are said to be a Tauberian theorems for the Np,qn,m,gCn,m,g1,1,1 summability method.

Next, we will find some conditions under which the converse implication holds, for defined convergence. Exactly, we will prove under which conditions statistical convergence of sequences xn,m,g, follows from statistically Nörlund-Cesáro summability method.

A sequence xn,m,g is weighted Np,qn,m,gCn,m,g1,1,1-statistically convergent to L if for every ε>0,


And we say that the sequence xn,m,g is statistically summable to L by the weighted summability method Np,qn,m,gCn,m,g1,1,1 if stlimn,m,gNp,qn,m,gCn,m,g1,1,1=L. We will denote by Np,qn,m,gCn,m,g1,1,1st the set of all sequences which are statistically summable, by the weighted summability method Np,qn,m,gCn,m,g1,1,1.

Theorem 1.1 Let x=xn,m,g be a sequence Np,qn,m,gCn,m,g1,1,1 summable to L, then the sequence x=xn,m,g is Np,qn,m,gCn,m,g1,1,1-statistically convergent to L, but not conversely.

Proof: The first part of the proof is obvious. To prove the second part, we will show the following example:

Let us define


and pn,m,g=1=qn,m,g. Under this conditions we obtain,


On the other hand, for i=n2, j=m2 and k=g2, we have


From last relation follows that x=xn,m,g is not Np,qn,m,gCn,m,g1,1,1 summable to 0.

Theorem 1.2 Let x=xn,m,g be a sequence statistically convergent to L and xn,m,gLM for every nmg××. Then, it converges Np,qn,m,gCn,m,g1,1,1-statistically to L.

Proof: From the fact that xn,m,g converges statistically to L, we have


We will denote Bε=ijknmg:xi,j,kLε and B¯ε=ijknmg:xi,j,kLε. Then,


for some constant C2.

Converse of Theorem 1.2 is not true as can be seen in the following example.

Consider that pn,m,g=n+1m+1g+1, qn,m,g=1 for some nmg0×0×0 and define the sequence x=xn,m,g as follows:


Under this conditions, after some basic calculations we get that x=xn,m,g is Np,qn,m,gCn,m,g1,1,1-summable to 1. Therefore, by Theorem 1.2, x=xn,m,g is Np,qn,m,gCn,m,g1,1,1-statistically convergent. On the other hand, the sequences p2;p=2,3,, t2;t=2,3, and o2;o=2,3, have natural density zero and it is clear that st-liminfn,m,gxn,m,g=0 and st-limsupn,m,gxn,m,g=1. Hence, xi,j,k is not statistically convergent.


2. Tauberian theorems under Np,qn,m,gCn,m,g1,1,1-statistically convergence

In this section, we show the results that we obtained. Throughout this paper, Rλn,m,g and Rλn,λm,λg will have the same meaning.

Consider that st-limi,j,kxi,j,k=L; xn,m,g is Np,qn,m,gCn,m,g1,1,1-statistically convergent and (13) satisfies, then for every t>1, is valid the following relation


and in case where 0<t<1,


The condition given by relation (13) is equivalent to the condition


Proof: Suppose that relation (13) is valid, 0<λ<1, w=λn=λn, e=λm=λm and r=λg=λg, nmg××. Then, it follows that


From above relation and definition of sequences pn,m,g and qn,m,g, we have


Conversely, suppose that (9) is valid. Now, let λ>1 be given and let λ1,λ2,λ3 be chosen such that 1<λ1,λ2,λ3<λ. Set w=λn=λn, e=λm=λm and r=λg=λg. From 0<1λ<1λ1,1λ2,1λ3<1, it follows that


provided λ1,λ2,λ3λ1n,λ1m,λ1g, which is a case where if n, m and g are large enough. Under this condition, we obtain


Consider that (13) is satisfied and let x=xi,j,k be a sequence of complex numbers which is Np,qn,m,gCn,m,g1,1,1-statistically convergent to L. Then,




Proof: We begin proving the case (10), i.e. when λ>1. Then, we have


From (12), definition of the sequence qn,m,g and relation limsupn,m,gRλn,m,gRλn,m,gRn,m,g<, we get (10).

Prove of (11) is made similarly to the prove of (10).

In the following theorem, we characterize the converse implication when the statistically convergence follows from its Np,qn,m,gCn,m,g1,1,1− statistically convergence.

Theorem 1.3 Let pn,m,g and qn,m,g be two non-negative real sequences and


where λn,m,g=λnλmλg=λnλmλg denotes the integral part of λnλmλg for every nmg××, and let xn,m,g be a sequence of real numbers which is Np,qn,m,gCn,m,g1,1,1-statistically convergent to a finite number L. Then, xn,m,g is st-convergent to the same number L if and only if the following two conditions hold




Proof: Necessity: Suppose that limn,m,gxn,m,g=L and (13) holds. By Proposition 2, we have


for every λ>1. In case where 0<λ<1, we have that


Sufficiency: Consider that (14) and ((15) are satisfied. In what follows, we will prove that limn,m,gxn,m,g=L. Given any ε>0, by (14) we can choose λ1>0 such that


where λn1=λn1, λm1=λm1 and λg1=λg1. By the assumed summability Np,qn,m,gCn,m,g1,1,1 of xn,m,g, Proposition 2 and (16), we have


for any λ>1.

On the other hand, if 0<λ<1, for every ε>0, we can choose 0<λ2<1 such that


where λn2=λn2, λm2=λm2 and λg2=λg2. By the assumed summability Np,qn,m,gCn,m,g1,1,1 of xn,m,g, Proposition 2 and (18), we have


for any 0<λ<1.

Since ε>0 is arbitrary, combining (17) and (19), we obtain


In the following theorem, we will consider the case where x=xn,m,g is a sequence of complex numbers.

Theorem 1.4 Let (13) be satisfied and let xn,m,g be a sequence of complex numbers which is Np,qn,m,gCn,m,g1,1,1-statistically convergent to a finite number L. Then, xn,m,g is convergent to the same number L if and only if the following two conditions hold




Proof: Necessity: If both (2) and (6) hold, then Proposition 2 yields (20) for every λ>1 and (21) for every 0<λ<1.

Sufficiency: Suppose that (2), (13) and one of the conditions (20) and (21) are satisfied. For any given ε>0, there exists λ1>0 such that


where λn1=λn1, λm1=λm1 and λg1=λg1. Taking into account fact that xn,m,g is Np,qn,m,gCn,m,g1,1,1 summbale to L and Proposition 2, we have the following estimation


For a given ε>0, there exists λ2>0 such that


where λn2=λn2, λm2=λm2 and λg2=λg2. Taking into account fact that xn,m,g is Np,qn,m,gCn,m,g1,1,1 summbale to L and Proposition 2, we obtain the following


Since ε>0 in either case, we get


3. Conclusion

In this paper, we have defined and proved new Tauberian theorems under triple statistically Nörlund-Cesáro summability, as a consequence of results showed in 2, some theorems, lemmas and corollaries can be defined and proved similarly by using 1,0,0, 0,1,0, and 0,0,1 method of summability. It is well know that Tauberian theorems for single sequences of single variable have been achieved a high degree of development; however, it is still in its infancy for triple sequences. For that reason, the results established in this paper can be extended and studied in some inclusion, Tauberian type theorems and Tauberian convexity type for certain families of generalized Nörlund.


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Written By

Carlos Granados

Reviewed: 28 June 2022 Published: 04 August 2022