# The riesz convergence and riesz core of double sequences

## Abstract

In this article, we have introduced the Riesz convergence and Riesz core of double sequences and determined the necessary and sufficient conditions on a four-dimensional matrix A to yield P R - core{Ax} P - core{x} and P R - core{Ax} st2 - core{x} for all $x\in {\ell }_{\infty }^{2}$.

Mathematics Subject Classification 2000: 40C05; 40J05; 46A45.

## 1. Introduction

A double sequence $x={\left[{x}_{jk}\right]}_{j,k=0}^{\infty }$ is said to be convergent in the Pringsheim sense or P-convergent if for every ε > 0 there exists an N such that | x jk - ℓ | < ε whenever j, k > N, . In this case, we write P-lim x = ℓ. By c2, we mean the space of all P-convergent sequences.

A double sequence x is bounded if

$∥x∥=\underset{j,k\ge 0}{\text{sup}}\left|{x}_{jk}\right|<\infty .$

By ${\ell }_{\infty }^{2}$ we denote the space of all bounded double sequences.

Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. So, we denote by ${c}_{2}^{\infty }$ the space of double sequences which are bounded and convergent.

Let E × and E(m, n) = {(j, k):j ≤ m,k ≤ n}. Then, the double natural density of E is defined by

${\delta }_{2}\left(E\right)=P-\underset{m,n}{\text{lim}}\frac{\left|E\left(m,n\right)\right|}{mn}$

if the limit on the right hand side exists; where the vertical bars denotes the cardinality of the set E(m,n).

A real double sequence x = [x jk ] is said to be statistical (or briefly st-) convergent  to the number L if for every ε > 0, the set {(j,k): |x jk - L| > ε} has double natural density zero. In this case, we write st2 - lim x = L. Let st2 be the space of all st-convergent double sequences. Clearly, a convergent double sequence is also st-convergent but the converse it is not true, in general. Also, note that a st-convergent double sequence need not be bounded. For example, consider the sequence x = [x jk ] defined by

(1.1)

Then, clearly st2-lim x = 1. Nevertheless x neither convergent nor bounded. The st2-lim sup and st2 - lim inf of a double sequence were introduced in  and also the statistical core of a double sequence was defined by the closed interval [st2 - lim sup, st2 - lim inf].

Let $A={\left[{a}_{jk}^{mn}\right]}_{j,k=0}^{\infty }$ be a four-dimensional infinite matrix of real numbers for all m,n = 0,1,.... The sums

${y}_{mn}=\sum _{j=0}^{\infty }\sum _{k=0}^{\infty }{a}_{jk}^{mn}{x}_{jk}$

are called the A- transforms of the double sequence x = [x jk ]. We say that a sequence x = [x jk ] is A-summable to the limit if the A- transform of x = [x jk ] exists for all m, n = 0,1,... and is convergent to in the Pringsheim sense, i.e.,

$\underset{p,q\to \infty }{\text{lim}}\sum _{j=0}^{p}\sum _{k=0}^{q}{a}_{jk}^{mn}{x}_{jk}={y}_{mn}$

and

$\underset{m,n\to \infty }{\text{lim}}{y}_{mn}=\ell .$

We say that a matrix A is bounded-regular if every bounded-convergent sequence x is A-summable to the same limit and the A-means are also bounded. The necessary and sufficient conditions for A to be bounded-regular or RH- regular are known (see [4, 5]).

A double sequence x = [x jk ] of real numbers is said to be Cesáro convergent to a number L if and only if there exists an L such that

$\underset{p,q\to \infty }{\text{lim}}\frac{1}{\left(p+1\right)\left(q+1\right)}\sum _{j=1}^{p}\sum _{k=1}^{q}{x}_{jk}^{mn}=L,$

and is denoted by C1 - lim x = L. We denote the space of all Cesáro convergent double sequences by C1. That is,

${C}_{1}=\left\{x\in {\ell }_{\infty }^{\mathsf{\text{2}}}:\exists L\in ℝ∍\phantom{\rule{1em}{0ex}}{C}_{1}-\text{lim}x=L\right\}.$

The concept core of single sequences (see ) was extended by Patterson  to the double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence x = [x jk ] as the closed interval [P - lim inf x,P - lim sup x]. Later this concept has been studied by many authors. For example we refer [2, 810].

Let

${C}_{1}^{*}\left(x\right)=\underset{p,q\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{\left(p+1\right)\left(q+1\right)}\sum _{j=0}^{p}\sum _{k=0}^{q}{x}_{jk}.$

The Cesáro core (or P C -core) of a real-valued bounded double sequence x = [x jk ] has been defined by the closed interval $\left[-{C}_{1}^{*}\left(-x\right),{C}_{1}^{*}\left(x\right)\right]$ in . Also; where an inequality related to the P C and P-cores has been investigated.

In this article we have introduced the Riesz convergence and Riesz core of a double sequence and also we have investigated some inequalities related to the P-, statistical and Riesz cores.

## 2. Main results

Definition 2.1. Let (q i ), (p j ) be sequences of non-negative numbers which are not all zero and Q m = q1 + q2 + · · · + q m , q1 > 0, P n = p1 + p2 + · · · + p n , p1 > 0. Then, the transformation given by

${t}_{mn}^{qp}\left(x\right)=\frac{1}{{Q}_{m}}\frac{1}{{P}_{n}}\sum _{i=1}^{m}\sum _{j=1}^{n}{q}_{i}{p}_{j}{x}_{ij}$

is called the Riesz mean of double sequence x = [x jk ].

Definition 2.2. If $P-\text{lim}\underset{mn}{\overset{qp}{t}}\left(x\right)=s$, s , then the sequence x = [x jk ] is said to be Riesz convergent to s.

If x = [x jk ] is Riesz convergent to s, then we write P R - lim x = s. In what follows ${c}_{R}^{2}$ will denote the set of all Riesz convergent sequences. Since a Riesz convergent double sequence need not be bounded, by ${c}_{R}^{2,\infty }$ we will denote the set of all bounded and Riesz convergent double sequences. ${c}_{0,R}^{2,\infty }$ will denote the set of all double sequences which bounded and Riesz convergent to zero.

Note that in the case q i = 1 for all i and p j = 1 for all j, the Riesz mean reduced to the Cesáro mean and the Riesz convergence is said to be Cesáro convergence.

Now, we will give some lemmas characterized some classes of matrices related to the ${c}_{R}^{2,\infty }$.

Lemma 2.3. A matrix $A=\left({a}_{jk}^{mn}\right)\in \left({\ell }_{\infty }^{2},{c}_{0,R}^{2,\infty }\right)$ if and only if

$∥A∥=\underset{mn}{\text{sup}}\sum _{jk}\left|{a}_{jk}^{mn}\right|<\infty ,$
(2.1)
$P-\underset{mn}{\text{lim}}\alpha \left(m,n,r,s,q,p\right)=0\left(r,s\in ℕ\right),$
(2.2)
$P-\underset{mn}{\text{lim}}\sum _{r}\left|\alpha \left(m,n,r,s,q,p\right)\right|=0\left(s\in ℕ\right),$
(2.3)
$P-\underset{mn}{\text{lim}}\sum _{s}\left|\alpha \left(m,n,r,s,q,p\right)\right|=0\left(r\in ℕ\right),$
(2.4)
$P-\underset{mn}{\text{lim}}\sum _{rs}\left|\alpha \left(m,n,r,s,q,p\right)\right|=0,$
(2.5)

where

$\alpha \left(m,n,r,s,q,p\right)=\frac{1}{{Q}_{r}}\frac{1}{{P}_{s}}\sum _{j=1}^{r}\sum _{k=1}^{s}{q}_{j}{p}_{k}{a}_{jk}^{mn}.$

Proof. Let $A=\left({a}_{jk}^{mn}\right)\in \left({\ell }_{\infty }^{2},{c}_{0,R}^{2,\infty }\right)$. This means that Ax exists for all $x=\left[{x}_{jk}\right]\in {\ell }_{\infty }^{2}$ and $Ax\in {c}_{0,R}^{2,\infty }$ which implies (2.1). Let us define a sequence y = [y rs ] by

${y}_{rs}=\left\{\begin{array}{c}\begin{array}{c}\mathsf{\text{sgn}}\phantom{\rule{1em}{0ex}}\alpha \left({m}_{i},{n}_{j},r,s,q,p\right),\phantom{\rule{1em}{0ex}}{r}_{i-1}

Then, the necessity of (2.5) follows from $P-\text{lim}\underset{rs}{\overset{qp}{t}}\left(Ax\right)$.

It is known by the assumption that

$P-\text{lim}\sum _{r,s}\alpha \left(m,n,r,s,q,p\right){x}_{jk}=0.$

So; if we define the sequences ${e}_{ij}^{rs},{e}^{r},{e}^{s}$ as follows

${e}_{ij}^{rs}=\left\{\begin{array}{c}1,\phantom{\rule{1em}{0ex}}\left(j,k\right)=\left(r,s\right)\\ 0\phantom{\rule{1em}{0ex}}\mathsf{\text{otherwise,}}\end{array}\right\$

er= Σ s ers(s ) and es= Σ r ers(r ), then the necessity of (2.2), (2.3), and (2.4) follows from $P-\text{lim}\underset{rs}{\overset{qp}{t}}\left(A{e}^{rs}\right),\phantom{\rule{2.77695pt}{0ex}}P-\text{lim}\underset{rs}{\overset{qp}{t}}\left(A{e}^{r}\right)$ and $P-\text{lim}\underset{rs}{\overset{qp}{t}}\left(A{e}^{s}\right)$, respectively.

Since the proof of the sufficiency part is routine, we omit the details.

Lemma 2.4. A matrix $A=\left({a}_{jk}^{mn}\right)\in {\left({c}_{2}^{\infty },{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$ if and only if (2.1)-(2.4) hold and

$P-\underset{mn}{\text{lim}}\sum _{rs}\left|\alpha \left(m,n,r,s,q,p\right)\right|=1.$
(2.6)

Proof. The necessity of the conditions can be shown by the same way used in the proof of Lemma 2.3.

For the sufficiency let the conditions hold and x = $\left[{x}_{jk}\right]\in {c}_{2}^{\infty }$ with P - lim x jk = L, (say). Then, there exists an N > 0 such that |x jk | <|L| + ε for every whenever j, k > N. Now; let us write

$\begin{array}{ll}\hfill \sum _{rs}\alpha \left(m,n,r,s,q,p\right){x}_{rs}=\sum _{r=0}^{N}\sum _{s=0}^{N}\alpha \left(m,n,r,s,q,p\right){x}_{rs}& +\sum _{r=N+1}^{\infty }\sum _{s=0}^{N-1}\alpha \left(m,n,r,s,q,p\right){x}_{rs}\phantom{\rule{2em}{0ex}}\\ +\sum _{r=0}^{N-1}\sum _{s=N+1}^{\infty }\alpha \left(m,n,r,s,q,p\right){x}_{rs}\phantom{\rule{2em}{0ex}}\\ +\sum _{r=N+1}^{\infty }\sum _{s=N+1}^{\infty }\alpha \left(m,n,r,s,q,p\right){x}_{rs}\phantom{\rule{2em}{0ex}}\end{array}$

which implies that

$\begin{array}{ll}\hfill \left|\sum _{rs}\alpha \left(m,n,r,s,q,p\right){x}_{rs}\right|=∥x∥\sum _{r=0}^{N}\sum _{s=0}^{N}\left|\alpha \left(m,n,r,s,q,p\right)\right|& +∥x∥\sum _{r=N+1}^{\infty }\sum _{s=0}^{N-1}\left|\alpha \left(m,n,r,s,q,p\right)\right|\phantom{\rule{2em}{0ex}}\\ +∥x∥\sum _{r=0}^{N-1}\sum _{s=N+1}^{\infty }\left|\alpha \left(m,n,r,s,q,p\right)\right|\phantom{\rule{2em}{0ex}}\\ +\left(\left|L\right|+\epsilon \right)\left|\sum _{r=N+1}^{\infty }\sum _{s=N+1}^{\infty }\alpha \left(m,n,r,s,q,p\right)\right|.\phantom{\rule{2em}{0ex}}\end{array}$

So, by letting m,n → ∞ under the light of the assumption, we get that $P-\text{lim}\underset{rs}{\overset{qp}{t}}\left(Ax\right)=L$.

This completes the proof.

Lemma 2.5. A matrix $A=\left({a}_{jk}^{mn}\right)\in {\left(s{t}_{2}\cap {\ell }_{\infty }^{2},{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$ if and only if

$A=\left({a}_{jk}^{mn}\right)\in {\left({c}_{2}^{\infty },{c}_{R}^{2,8}\right)}_{\mathsf{\text{reg}}}$
(2.7)
$P-\underset{mn}{\text{lim}}\sum _{r,s\in E}\left|\alpha \left(m,n,r,s,q,p\right)\right|=0$
(2.8)

for every E × with δ2(E) = 0.

Proof. If $A=\left({a}_{jk}^{mn}\right)\in {\left(s{t}_{2}\cap {\ell }_{\infty }^{2},{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$, the necessity of (2.7) follows from the fact that ${c}_{2}^{\infty }\subset s{t}_{2}\cap {\ell }_{\infty }^{2}$. For the necessity of the condition (2.8), let us choose a sequence z = [z rs ] by

${z}_{rs}=\left\{\begin{array}{c}{x}_{rs},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}r,s\in E\\ 0,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathsf{\text{otherwise,}}\end{array}\right\$

where $x=\left[{x}_{rs}\right]\in {\ell }_{\infty }^{2}$ and E × with δ2 (E) = 0. Then; it is easy to see that st2 - lim z = 0 and

${t}_{rs}^{qp}\left(Az\right)=\sum _{r,s\in E}\alpha \left(m,n,r,s,q,p\right){x}_{rs}.$

So; a matrix $B=\left[{b}_{rs}^{mn}\right]$ defined by

${b}_{rs}^{mn}=\left\{\begin{array}{c}\alpha \left(m,n,r,s,q,p\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}r,s\in E\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}0,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathsf{\text{otherwise}},\end{array}\right\$

for every q,p is in the class $\left({\ell }_{\infty }^{2},{c}_{0,R}^{2,\infty }\right)$. Therefore, the necessity of (2.8) follows from the condition (2.5) of Lemma 2.3.

For the converse take a sequence $x=\left[{x}_{rs}\right]\in s{t}_{2}\cap {\ell }_{\infty }^{2}$ with st2 - lim x = l. Then; it is known that δ2 = δ2({(r, s): |x rs - l| ≥ ε}) = 0 and |x rs - l| < ε whenever r,s E. Now, write

$\sum _{r,s}\alpha \left(m,n,r,s,q,p\right){x}_{rs}=\sum _{r,s}\alpha \left(m,n,r,s,q,p\right)\left({x}_{rs}-1\right)+l\sum _{r,s}\alpha \left(m,n,r,s,q,p\right).$
(2.9)

The inequality

$\begin{array}{ll}\hfill \left|\sum _{r,s}\alpha \left(m,n,r,s,q,p\right)\left({x}_{rs}-l\right)\right|& =\left|\sum _{r,s\in E}\alpha \left(m,n,r,s,q,p\right)\left({x}_{rs}-1\right)+\sum _{r,s\notin E}\alpha \left(m,n,r,s,q,p\right)\left({x}_{rs}-1\right)\right|\phantom{\rule{2em}{0ex}}\\ \le ∥{x}_{rs}-l∥\sum _{r,s\in E}\left|\alpha \left(m,n,r,s,q,p\right)\right|+\epsilon ∥A∥\phantom{\rule{2em}{0ex}}\end{array}$

and condition (2.8) implies that

$P-\underset{mn}{\text{lim}}\sum _{r,s}\alpha \left(m,n,r,s,q,p\right)\left({x}_{rs}-l\right)=0.$

So; by letting m, n → ∞ in (2.9) we have $P-\text{lim}\underset{rs}{\overset{qp}{t}}\left(Ax\right)=l$ and this completes the proof.

Definition 2.6. The Riesz core (or P R -core) of a double sequence x = [x jk ] is the closed interval $\left[P-\mathsf{\text{lim}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{in}}{\mathsf{\text{f}}}_{m,n}{t}_{mn}^{qp}\left(x\right),P-\mathsf{\text{lim}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{su}}{\mathsf{\text{p}}}_{m,n}{t}_{mn}^{qp}\left(x\right)\right].$.

Note that in the case q i = 1 for all i and p j = 1 for all j, Riesz core is reduced to the Cesáro core, .

Now; we are ready to give some inequalities related to the P-, P R - and st2-core of double sequences.

Theorem 2.7. Let A < ∞. Then,

$P-\text{lim}\text{sup}{t}_{rs}^{qp}\left(Ax\right)\le P-\text{lim}\text{sup}\left(x\right),$
(2.10)

for all $x\in {\ell }_{\infty }^{2}$ if and only if $A\in {\left({c}_{2}^{\infty },{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$ and

$P-\underset{m,n}{\text{lim}}\sum _{r,s}\left|\alpha \left(m,n,r,s,q,p\right)\right|=1.$
(2.11)

Proof. Let (2.10) holds for all $x\in {\ell }_{\infty }^{2}$. Then, it is easy to get that

Since - P - lim sup(-x) = P - lim inf(x) and $-P-\text{lim}\text{sup}\underset{rs}{\overset{qp}{t}}\left(-Ax\right)=P-\text{lim}\text{inf}\underset{rs}{\overset{qp}{t}}\left(Ax\right)$, by choosing $x\in {c}_{2}^{\infty }$, we reach that $P-\text{lim}\underset{rs}{\overset{qp}{t}}\left(Ax\right)=P-\text{lim}\left(x\right)$. Since x is arbitrary, this means that $A\in {\left({c}_{2}^{\infty },{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$.

By Lemma 3.1 of Patterson , there exists a $y\in {\ell }_{\infty }^{2}$ with || y || ≤ 1 such that

$P-\text{lim}\text{sup}\underset{rs}{\overset{qp}{t}}\left(Ay\right)=P-\text{lim}\text{sup}\sum _{r,s}\left|\alpha \left(m,n,r,s,q,p\right)\right|.$

So; we have from assumption that

$P-\text{lim}\text{sup}\sum _{r,s}\left|\alpha \left(m,n,r,s,q,p\right)\right|=p-\text{lim}\text{sup}\underset{rs}{\overset{qp}{t}}\left(Ay\right)\le P-\text{lim}\text{sup}\left(y\right)\le ∥y∥\le 1.$
(2.12)

By the same way, one can see that

$P-\text{lim}\text{inf}\sum _{r,s}\left|\alpha \left(m,n,r,s,q,p\right)\right|\ge 1.$
(2.13)

Therefore, by combining the inequalities (2.12) and (2.13), we obtain the necessity of (2.11).

Conversely; suppose that $A\in {\left({c}_{2}^{\infty },{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$ and (2.11) holds. For any arbitrary bounded sequence x = [x rs ], there exists M, N > 0 such that x rs ≤ P - lim sup x + ε whenever r > M, s > N. Now, we can write the following inequality,

Using the conditions characterized the class ${\left({c}_{2}^{\infty },{c}_{R}^{2,\infty }\right)}_{\text{reg}}$ and (2.11), we reach that $P-\text{lim}\text{sup}\underset{rs}{\overset{qp}{t}}\left(Ax\right)\le P-\text{lim}\text{sup}\left(x\right)$ and this completes the proof of the theorem.

Theorem 2.8. Let A < ∞. Then,

$P-\text{lim}\text{sup}\underset{rs}{\overset{qp}{t}}\left(Ax\right)\le s{t}_{2}-\text{lim}\text{sup}\left(x\right),$
(2.14)

for all x $x\in {\ell }_{\infty }^{2}$ if and only if $A\in {\left(s{t}_{2}\cap {\ell }_{\infty }^{2},{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$ and (2.11) holds.

Proof. Let (2.14) holds for all $x\in {\ell }_{\infty }^{2}$. Then, by the same argument used in Theorem 2.7, one can see that $A\in {\left(s{t}_{2}\cap {\ell }_{\infty }^{2},{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$. On the other since st2 - lim sup(x) ≤ P - lim sup(x) for all $x\in {\ell }_{\infty }^{2}$, the necessity of (2.11) follows from Theorem 2.7.

For the converse suppose that $A\in {\left(s{t}_{2}\cap {\ell }_{\infty }^{2},{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$ and (2.11) holds. If $x=\left[{x}_{rs}\right]\in {\ell }_{\infty }^{2},$, it is known that for every ε > 0,

${\delta }_{2}\left(E\right)={\delta }_{2}\left(\left\{\left(r,s\right):{x}_{rs}>s{t}_{2}-\text{lim}\text{sup}\left(x\right)+\epsilon \right\}\right)=0$

and x rs ≤ st2 - lim sup(x) + ε whenever r, s E. Taking this knowledge in the mind, let us write

So, the conditions characterized the class ${\left(s{t}_{2}\cap {\ell }_{\infty }^{2},{c}_{R}^{2,\infty }\right)}_{\mathsf{\text{reg}}}$ and (2.11) imply that $P-\text{lim}\text{sup}\underset{rs}{\overset{qp}{t}}\left(Ax\right)\le s{t}_{2}-\text{lim}\text{sup}\left(x\right)+\epsilon$. Since ε was arbitrary, this completes the proof.

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Correspondence to Abdullah M Alotaibi.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

AMA designed the problems and carried out the proof of the Lemmas. CÇ defined the Riesz core and gave the proof of the theorems. All authors read and approved the final manuscript.

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Alotaibi, A.M., Çakan, C. The riesz convergence and riesz core of double sequences. J Inequal Appl 2012, 56 (2012). https://doi.org/10.1186/1029-242X-2012-56

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• DOI: https://doi.org/10.1186/1029-242X-2012-56

### Keywords

• double sequences
• four dimensional matrices
• Riesz mean
• core theorems 