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# Weighted almost convergence and related infinite matrices

*Journal of Inequalities and Applications*
**volume 2018**, Article number: 15 (2018)

## Abstract

The purpose of this paper is to introduce the notion of weighted almost convergence of a sequence and prove that this sequence endowed with the sup-norm \({\Vert \cdot \Vert } _{\infty}\) is a BK-space. We also define the notions of weighted almost conservative and regular matrices and obtain necessary and sufficient conditions for these matrix classes. Moreover, we define a weighted almost *A*-summable sequence and prove the related interesting result.

## Introduction and preliminaries

Let *ω* denote the space of all complex sequences \(s=(s_{j})_{j=0}^{\infty}\) (or simply write \(s=(s_{j})\)). Any vector subspace of *ω* is called a sequence space. By \(\mathbb {N}\) we denote the set of natural numbers, and by \(\mathbb {R}\) the set of real numbers. We use the standard notation \(\ell_{\infty}\), *c* and \(c_{0}\) to denote the sets of all bounded, convergent and null sequences of real numbers, respectively, where each of the sets is a Banach space with the sup-norm \(\Vert . \Vert _{\infty}\) defined by \(\Vert s \Vert _{\infty}=\sup_{j\in\mathbb {N}} \vert s_{j} \vert \). We write the space \(\ell_{p}\) of all absolutely *p*-summable series by

Clearly, \(\ell_{p}\) is a Banach space with the following norm:

For \(p=1\), we obtain the set \(l_{1}\) of all absolutely summable sequences. For any sequence \(s=(s_{j})\), let \(s^{[n]}=\sum_{j=0}^{n}s_{j}e_{j}\) be its *n*-section, where \(e_{j}\) is the sequence with 1 in place *j* and 0 elsewhere and \(e=(1,1,1,\dots)\).

A sequence space *X* is called a BK-space if it is a Banach space with continuous coordinates \(p_{j}:X\to\mathbb {C}\), the set of complex fields, and \(p_{j}(s)=s_{j}\) for all \(s=(s_{j})\in X\) and every \(j\in \mathbb {N}\). A BK-space \(X\supset\psi\), the set of all finite sequences that terminate in zeros, is said to have AK if every sequence \(s=(s_{j})\in X\) has a unique representation \(s=\sum_{j=0}^{\infty }s_{j}e_{j}\).

Let *X* and *Y* be two sequence spaces, and let \(A=(a_{n,k})\) be an infinite matrix. If, for each \(s=(s_{k})\) in *X*, the series

converges for each \(n\in\mathbb {N}\) and the sequence \(As=(A_{n}s)\) belongs to *Y*, then we say that matrix *A* maps *X* into *Y*. By the symbol \((X,Y)\) we denote the set of all such matrices which map *X* into *Y*. The series in (1) is called *A*-*transform* of *s* whenever the series converges for \(n=0,1,\ldots \) . We say that \(s=(s_{k})\) is *A*-*summable* to the limit *λ* if \(A_{n}s\) converges to *λ* (\(n\to\infty\)).

The sequence \(s=(s_{k})\) of \(\ell_{\infty}\) is said to be almost convergent, denoted by *f*, if all of its Banach limits [1] are equal. We denote such a class by the symbol *f*, and one writes \(f\mbox{-}\lim s =\lambda\) if *λ* is the common value of all Banach limits of the sequence \(s=(s_{k})\). For a bounded sequence \(s=(s_{k})\), Lorentz [2] proved that \(f\mbox{-}\lim s =\lambda \) if and only if

uniformly in *m*. This notion was later used to (i) define and study conservative and regular matrices [3]; (ii) introduce related sequence spaces derived by the domain of matrices [4–6]; (iii) study some related matrix transformations [7–9]; (iv) define related sequence spaces derived as the domain of the generalized weighted mean and determine duals of these spaces [10, 11]. As an extension of the notion of almost convergence, Kayaduman and Şengönül [12, 13] defined Cesàro and Riesz almost convergence and established related core theorems. The almost strongly regular matrices for single sequences were introduced and characterized [14], and for double sequences they were studied by Mursaleen [15] (also refer to [16–19]). As an application of almost convergence, Mohiuddine [20] proved a Korovkin-type approximation theorem for a sequence of linear positive operators and also obtained some of its generalizations. Başar and Kirişçi [21] determined the duals of the sequence space *f* and other related spaces/series and investigated some useful characterizations.

We now recall the following result.

### Lemma 1.1

([22])

*Let*
*X*
*and*
*Y*
*be BK*-*spaces*. (i) *Then*
\((X,Y)\subset B(X,Y)\), *that is*, *every*
\(A\in(X,Y)\)
*defines an operator*
\({\mathcal{L}}_{A}\in B(X,Y)\)
*by*
\({\mathcal{L}}_{A}(x)=Ax\)
*for all*
\(x\in X\), *where*
\(B(X,Y)\)
*denotes the set of all bounded linear operators from*
*X*
*into*
*Y*. (ii) *Then*
\(A\in(X,\ell_{\infty})\)
*if and only if*
\(\Vert A \Vert _{(X,\ell_{\infty})}=\sup_{n} \Vert A_{n} \Vert _{X}<\infty\). *Moreover*, *if*
\(A\in(X,\ell _{\infty})\), *then*
\(\Vert {\mathcal{L}}_{A} \Vert = \Vert A \Vert _{(X,\ell_{\infty})}\).

## Weighted almost convergence

### Definition 2.1

Let \(t=(t_{k})\) be a given sequence of nonnegative numbers such that \(\liminf_{k} t_{k}>0\) and \(T_{m}=\sum_{k=0}^{m-1}t_{k}\neq0\) for all \(m\geq1\). Then the bounded sequence \(s=(s_{k})\) of real or complex numbers is said to be *weighted almost convergent*, shortly \(f(\bar{N})\)-convergent, to *λ* if and only if

We shall use the notation \(f(\bar{N})\) for the space of all sequences which are \(f(\bar{N})\)-convergent, that is,

We remark that if we take \(t_{k}=1\) for all *k*, then (2) is reduced to the notion of almost convergence introduced by Lorentz [2]. Clearly, a convergent sequence is \(f(\bar{N})\)-convergent to the same limit, but its converse is not always true.

### Example 2.2

Consider a sequence \(s=(s_{k})\) defined by \(s_{k}=1\) if *k* is odd and 0 for even *k*. Also, let \(t_{k}=1\) for all *k*. Then we see that \(s=(s_{k})\) is \(f(\bar{N})\)-convergent to \(1/2\) but not convergent.

### Definition 2.3

The matrix *A* (or a matrix map *A*) is said to be *weighted almost conservative* if \(As\in f({\bar {N}})\) for all \(s=(s_{k})\in c\). One denotes this by \(A\in(c,f({\bar {N}}))\). If \(A\in(c,f({\bar{N}}))\) with \(f({\bar{N}})\mbox{-}\lim As=\lim s\), then we say that *A* is *weighted almost regular matrix*; one denotes such matrices by \(A\in(c,f({\bar{N}}))_{R}\).

### Theorem 2.4

*The space*
\(f(\bar{N})\)
*of weighted almost convergence endowed with the norm*
\({\Vert \cdot \Vert } _{\infty}\)
*is a BK*-*space*.

### Proof

To prove our results, first we have to prove that \(f(\bar{N})\) is a Banach space normed by

where

It is easy to verify that (3) defines a norm on \(f(\bar{N})\). We have to show that \(f(\bar{N})\) is complete. For this, we need to show that every Cauchy sequence in \(f(\bar{N})\) converges to some number in \(f(\bar{N})\). Let \((s^{k})\) be a Cauchy sequence in \(f(\bar{N})\). Then \((s_{j}^{k})\) is a Cauchy sequence in \(\mathbb {R}\) (for each \(j=1,2,\dots\)). By using the notion of the norm of \(f(\bar{N})\), it is easy to see that \((s^{k})\to s\). We have only to show that \(s\in f(\bar{N})\).

Let \(\epsilon>0\) be given. Since \((s^{k})\) is a Cauchy sequence in \(f(\bar{N})\), there exists \(M\in\mathbb {N}\) (depending on *ϵ*) such that

which yields

Therefore we have \(\vert \Psi(s^{k}-s^{i}) \vert <\epsilon/3\). Taking the limit as \(m\to\infty\) gives that \(\vert \lambda ^{k}-\lambda^{i} \vert <\epsilon/3\) for each *m*, *r* and \(k,i>M\), where \(\lambda^{k}=f(\bar{N})\mbox{-}\lim_{m} s^{k}\) and \(\lambda ^{i}=f(\bar{N})\mbox{-}\lim_{m} s^{i}\). Let \(\lambda=\lim_{r\to \infty}\lambda^{i}\). Letting \(i\to\infty\), one obtains

for each *m*, *r* and \(k>M\). Now, for fixed *k*, the above inequality holds. Since \(s^{k}\in f(\bar{N})\), for fixed *k*, we get

For given \(\epsilon>0\), there exists positive integers \(M_{0}\) (independent of *r*, but dependent upon *ϵ*) such that

for \(m>M_{0}\) and for all *r*. It follows from (4) and (5) that

This proves that \(f(\bar{N})\) is a Banach space normed by (3).

Since \(c\subset f(\bar{N})\subset l_{\infty}\), there exist positive real numbers *α* and *β* with \(\alpha<\beta\) such that \(\alpha \Vert s \Vert _{\infty}\leq \Vert s \Vert _{f(\bar{N})}\leq\beta \Vert s \Vert _{\infty}\). That is to say, two norms \({\Vert \cdot \Vert } _{\infty}\) and \({\Vert \cdot \Vert } _{f(\bar{N})}\) are equivalent. It is well known that the spaces *c* and \(l_{\infty}\) endowed with the norm \({\Vert \cdot \Vert } _{\infty}\) are BK-spaces, and hence the space \(f(\bar{N})\) endowed with the norm \({\Vert \cdot \Vert } _{\infty}\) is also a BK-space. □

We prove the following characterization of weighted almost conservative matrices.

### Theorem 2.5

*The matrix*
\(A=(a_{n,k})\)
*is weighted almost conservative*, *that is*, \(A\in(c,f({\bar{N}}))\)
*if and only if*

### Proof

Necessity. Let \(A\in(c,f({\bar{N}}))\). Since the sequences *e* and \(e_{k}\) both are convergent, so *A*-transforms of the sequences \(e_{k}\) and *e* belong to \(f({\bar{N}})\) and exist uniformly in *r*. It follows that (7) and (8) are valid. Let *r* be any nonnegative integer. One writes

where

It follows that \(\alpha_{n}\in c'\) for all \(n\in\mathbb {N}_{0}:=\mathbb {N}\cup\{0\}\) and this yields \(\Phi_{mr}\in c'\) (\(m\geq 1\)). Since \(A\in(c,f({\bar{N}}))\),

It is clear that \((\Phi_{mr}(s))\) is bounded for \(s=(s_{k})\in c\) and fixed *r*. Hence, by the uniform boundedness principle, \(( \Vert \Phi_{mr} \Vert )\) is bounded. For each \(p\in\mathbb {Z}^{+}\) (the positive integers), the sequence \(x=(x_{k})\) is defined by

Then a sequence \(x\in c\), \(\Vert x \Vert =1\) and

Therefore, we obtain

Equations (9) and (10) give that

it follows that (6) is valid.

Sufficiency. Let conditions (6)-(8) hold. Let *r* be any nonnegative integer, and let \(s_{k}\in c\). Then

which gives

It follows from hypothesis (6) that \(\vert \Phi_{mr}(s) \vert \leq B_{r} \Vert s \Vert \), where \(B_{r}\) is a constant independent of *r*. Thus we have \(\Phi_{mr}\in c^{\prime}\) for each \(m\geq1\), which gives that a sequence \(( \Vert \Phi_{mr} \Vert )\) is bounded for each nonnegative integer *r*. Hypotheses (7) and (8) imply that the limit of \(\Phi_{mr}(e_{k})\) and \(\Phi_{mr}(e)\) must exist for all nonnegative integers *k* and *r*. Since \(\{e,e_{0},e_{1},\dots\}\) is a fundamental set in *c*, it follows from [23, p. 252] that \(\lim_{m}\Phi_{mr}(s)=\Phi _{r}(s)\) exists and \(\Phi_{r}\in c^{\prime}\). Therefore \(\Phi_{r}\) has the following form (see [23, p. 205]):

where \(\xi=\lim s_{k}\). From (7) and (8), we see that \(\Phi_{r}(e_{k})=\lambda_{k}\) for a nonnegative integer *k* and \(\Phi_{r}(e)=\lambda\). Therefore, for each \(s\in c\) and a nonnegative integer *r*, we have

with the following expression:

Since \(\Phi_{mr}\in c^{\prime}\), so it has the representation

We observe from (11) and (12) that the convergence of \(\Phi_{mr}(s)\) to \(\Phi(s)\) is uniform since \(\lim_{m\to\infty }\Phi_{mr}(e_{k})=\lambda_{k}\) and \(\lim_{m\to\infty}\Phi _{mr}(e)=\lambda\) uniformly in *r*. Hence, *A* is a weighted almost conservative matrix. □

In the following theorem, we obtain the characterization of weighted almost regular matrices.

### Theorem 2.6

*The matrix*
\(A\in(c,f({\bar{N}}))_{R}\)
*if and only if*

### Proof

Necessity. Let \(A\in(c,f({\bar{N}}))_{R}\). We see that condition (13) holds by using the fact that *A* is also weighted almost conservative. Take \(e_{k},e\in c\). Then *A*-transforms of the sequences \(e_{k}\) and *e* are weighted almost convergent to 0 and 1, respectively, since \(e_{k}\to0\) and \(e\to1\). Hence \(e_{k}\in c\) gives condition (14) and \(e\in c\) proves the validity of (15).

Sufficiency. Let conditions (13)-(15) hold. It is easy to see that *A* is weighted almost conservative. So, for each \((s_{k})\in c\), \(\lim_{m\to\infty}\Phi_{mr}(s)=\Phi(s)\) uniformly in *r*. Thus we obtain from (11) and our hypotheses (13)-(15) that \(\Phi(s)=\xi=\lim s_{k}\). This yields *A* is weighted almost regular. □

We now obtain necessary and sufficient conditions for the matrix *A* which transform the absolutely convergent series into the space of weighted almost convergence.

### Theorem 2.7

*The matrix*
\(A\in(l_{1},f({\bar{N}}))\)
*if and only if*

### Proof

Necessity. Let \(A\in(l_{1},f({\bar{N}}))\). Condition (17) follows since \(e_{k}\in l_{1}\). Let \(\Phi_{mr}\) be a continuous linear functional on \(l_{1}\) defined by

Then we have

which yields

For any fixed \(k\in\mathbb {N}_{0}\), we define a sequence \(s=(s_{j})\) by

Then we have \(\Vert s \Vert _{1}=1\) and

so

We obtain from (18) and (19) that

Since \(A\in(l_{1},f({\bar{N}}))\), for any \(s\in l_{1}\), we have

By using the uniform boundedness theorem, Equation (20) becomes

This proves the validity of (16).

Sufficiency. Let conditions (16) and (17) hold, and let \(s=(s_{k})\in l_{1}\). In virtue of these conditions, we see that

it also converges absolutely. Furthermore, \(\frac{1}{T_{m}}\sum_{k=0}^{\infty}\sum_{n=r}^{r+m-1}t_{n}a_{n,k}s_{k}\) converges absolutely for each *m* and *r*.

Let \(\epsilon>0\) be given. Then there exists \(k_{0}\in\mathbb {N}\) such that

By condition (17), we can find some \(m_{0}\in\mathbb {N}\) such that

for all \(m>m_{0}\) uniformly in *r*. Now

for all \(m>m_{0}\) uniformly in *r*. By using Equations (22) and (23) and our hypotheses in the above inequality, we see that (21) holds, and hence the sufficiency part. □

### Theorem 2.8

*If the matrix*
*A*
*in*
\((l_{1},f({\bar {N}}))\), *then*
\(\Vert {\mathcal{L}}_{A} \Vert = \Vert A \Vert \).

### Proof

Let \(A\in(l_{1},f({\bar{N}}))\). Then we have

which gives \(\Vert {\mathcal{L}}_{A}(s) \Vert \leq \Vert A \Vert \Vert s \Vert _{1}\). This implies that \(\Vert {\mathcal{L}}_{A} \Vert \leq \Vert A \Vert \). Also, \({\mathcal{L}}_{A}\in B(l_{1},f({\bar{N}}))\) gives

Taking \(s=(e_{k})\) and using the fact that \(\Vert e_{k} \Vert _{1}=1\) ∀*k*, one obtains \(\Vert A \Vert \leq \Vert {\mathcal{L}}_{A} \Vert \). Hence we conclude that \(\Vert {\mathcal{L}}_{A} \Vert = \Vert A \Vert \). □

### Definition 2.9

Let \(t=(t_{k})_{k\in\mathbb {N}}\) be a given sequence of nonnegative numbers such that \(\liminf_{k} t_{k}>0\) and \(T_{m}=\sum_{k=0}^{m-1}t_{k}\neq0\) for all \(m\geq1\). A sequence \(s=(s_{k})\) is said to be *weighted almost*
*A-summable* to \(\lambda\in\mathbb {C}\) if the *A*-transform of sequence \(s=(s_{k})\) is weighted almost convergent to *λ*; equivalently, we can write

where

In the applications of summability theory to function theory, it is important to know the region in which \(S=(S_{k}(z))\), the sequence of partial sums of the geometric series is *A*-summable to \(\frac {1}{1-z}\) for a given matrix *A*. In the following theorem, we find the region in which *S* is weighted almost *A*-summable to \(\frac{1}{1-z}\).

### Theorem 2.10

*Let*
\(A=(a_{n,k})\)
*be a matrix such that* (15) *holds*. *The sequence*
\((S_{k}(z))\)
*is weighted almost*
*A*-*summable to*
\(\frac{1}{1-z}\)
*if and only if*
\(z\in R\), *where*

### Proof

One writes

Taking the limit as \(m\to\infty\) in the above equality and using condition (15), one obtains

if and only if \(z\in R\). This completes the proof. □

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## Acknowledgements

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-694-D1435). The authors, therefore, gratefully acknowledge the DSR technical and financial support.

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### Cite this article

Mohiuddine, S.A., Alotaibi, A. Weighted almost convergence and related infinite matrices.
*J Inequal Appl* **2018, **15 (2018). https://doi.org/10.1186/s13660-017-1600-z

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DOI: https://doi.org/10.1186/s13660-017-1600-z

### MSC

- 46A45
- 40G99
- 40C05

### Keywords

- almost convergence
- weighted almost convergence
- regular matrix