We define the notions of double statistically convergent and double lacunary statistically convergent sequences in locally solid Riesz space and establish some inclusion relations between them. We also prove an extension of a decomposition theorem in this setup. Further, we introduce the concepts of double θ-summable and double statistically lacunary summable in locally solid Riesz space and establish a relationship between these notions.

1. Introduction

Fast [1] and Steinhaus [2] independently introduced an extension of the usual concept of sequential limits which he called statistical convergence. Actually the idea of statistical convergence was formerly given under the name “almost convergence" by Zygmund in the first edition (Warsaw, 1935) of his celebrated monograph [3]. Schoenberg [4] and Šalát [5] gave some basic properties of statistical convergence. In 1985, Fridy [6] introduced the notion of statistically Cauchy sequence and proved that it is equivalent to the concept of statistical convergence. The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences/matrices through the concept of density. Later on it was further investigated by various authors in different frameworks (see [7–18]). Mursaleen and Edely [19] extended these concepts from single to double sequences by using two dimensional analogue of natural density of subsets of ℕ×ℕ and established relationship between statistical convergence and strongly Cesáro summable double sequences. Mohiuddine et al. [20] and Mursaleen and Mohiuddine [21] defined these notions for double sequences in fuzzy normed spaces and intuitionistic fuzzy normed spaces, respectively. Recently, Mohiuddine et al. [22] introduced these notions for double sequences in locally solid Riesz spaces and proved some interesting results. Fridy and Orhan [23] presented an interesting generalization of statistical convergence with the help of lacunary sequence and called it lacunary statistical convergence. Savaş and Patterson [24, 25] extended the notion of lacunary statistical convergence from single sequences to double sequences with the help of double lacunary density and proved some interesting results related to this concept. For more details related to the concept of lacunary statistical convergence for single and double sequences and applications to approximation theorems, we refer to [26–42].

On the other hand, the concept of Riesz space was introduced by Riesz [43]. Since then, with a view to utilize this concept in topology and analysis, many authors have extensively developed the theory of Riesz spaces along with their applications (e.g., [7, 22, 44, 45]).

2. Definitions and Notations

In this section, we recall some of the basic concepts related to the notions of statistical convergence and lacunary sequence which we will use throughout the paper.

Let E⊆ℕ. Then the natural density of E is denoted by δ(E) and is defined by(1)δ(E)=limn→∞1n|{k≤n:k∈E}|exists,
where the vertical bar denotes the cardinality of the respective set.

Definition 1 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

A sequence x=(xk) in a topological space X is said to be statistically convergent to ℓ if for every neighborhood V of ℓ(2)δ({k∈ℕ:xk∉V})=0.
In this case, we write S-limx=ℓ.

By a lacunary sequence θ=(kr), where k0=0, we will mean an increasing sequence of nonnegative integers with hr:kr-kr-1→∞ as r→∞. The intervals determined by θ will be denoted by Ir=(kr-1,kr] and the ratio kr/kr-1 will be defined by qr (see [46]).

Definition 2.

Let θ be a lacunary sequence and let Ir={k:kr-1<k≤kr}. Let K⊂ℕ. The number δθ(K) is called the lacunary density or θ-density of K if
(3)δθ(K)=limr1hr|{i∈Ir:i∈K}|exists.

The generalized lacunary mean is defined by
(4)tr(x)=1hr∑k∈Irxk.

Definition 3.

A sequence x=(xk) is said to be θ-summable to number ℓ if tr(x)→ℓ as r→∞. In this case we write that ℓ is the θ-limit of x. If θ=(2r), then θ-summable reduces to C1-summable (see [46]).

By the convergence of a double sequence we mean the convergence in the Pringsheim sense [47]. A double sequence x=(xk,l) has a Pringsheim limit L (denoted by P-limx=L) provided that given an ε>0 there exists an n∈ℕ such that |xk,l-L|<ε whenever k,l>n. We will describe such an x=(xk,l) more briefly as “P-convergent.”

Let K⊂ℕ×ℕ, and let K(m,n) denote the number of (i,j) in K such that i≤m and j≤n (see [19]). Then the lower natural density of K is defined by δ_2(K)=liminfm,n→∞(|K(m,n)|/mn). In case that the sequence (K(m,n)/mn) has a limit in Pringsheim’s sense, then we say that K has a double natural density and is defined by P-limm,n→∞(|K(m,n)|/mn)=δ2(K).

For example, let K={(i2,j2):i,j∈ℕ}. Then
(5)δ2(K)=P-limm,n→∞|K(m,n)|mn≤P-limm,n→∞mnmn=0;
that is, the set K has double natural density zero, while the set {(i,3j):i,j∈ℕ} has double natural density 1/3.

The double sequence θ¯=θr,s={(kr,ls)} is called double lacunary sequence if there exist two increasing sequences of integers such that (see [25])
(6)ko=0,hr=kr-kr-1⟶∞asr⟶∞,lo=0,hs¯=ls-ls-1⟶∞ass⟶∞.

Notations. kr,s=krls, hr,s=hrhs¯, and θr,s is determined by
(7)Ir,s={(k,l):kr-1<k≤kr,ls-1<l≤ls},qr=krkr-1,qs¯=lsls-1,qr,s=qrqs¯.

Definition 4 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

Let θ¯={(kr,ls)} be a double lacunary sequence. Let K⊆ℕ×ℕ. The number
(8)δθ¯(K)=P-limr,s1hr,s|{(i,j)∈Ir,s:(i,j)∈K}|
is said to be double lacunary density, that is, θ¯-density of K, provided the limit exists.

We define the generalized double lacunary mean by
(9)tr,s(x)=1hr,s∑(k,l)∈Ir,sxk,l.

3. Locally Solid Riesz Spaces

Let X be a real vector space and let ≤ be a partial order on this space. Then X is said to be an ordered vector space if it satisfies the following properties:

if x,y∈X and y≤x, then y+z≤x+z for each z∈X;

if x,y∈X and y≤x, then ay≤ax for each a≥0.

If, in addition, X is a lattice with respect to the partial order, then X is said to be a Riesz space (or a vector lattice) (see [45]).

For an element x of a Riesz space X, the positive part of x is defined by x+=x∨0¯=sup{x,0¯}, the negative part of x by x-=(-x)∨0¯, and the absolute value of x by |x|=x∨(-x), where 0¯ is the zero element of X.

A subset S of a Riesz space X is said to be solid if y∈S and |y|≤|x| implies x∈S.

A topological vector space (X,τ) is a vector space X which has a topology (linear) τ, such that the algebraic operations of addition and scalar multiplication in X are continuous. Continuity of addition means that the function f:X×X→X defined by f(x,y)=x+y is continuous on X×X, and continuity of scalar multiplication means that the function f:ℝ×X→X defined by f(a,x)=ax is continuous on ℝ×X.

Every linear topology τ on a vector space X has a base N for the neighborhoods of 0¯ satisfying the following properties.

Each Y∈N is a balanced set; that is, ax∈Y holds for all x∈Y and for every a∈ℝ with |a|≤1.

Each Y∈N is an absorbing set; that is, for every x∈X, there exists a>0 such that ax∈Y.

For each Y∈N there exists some E∈N with E+E⊆Y.

A linear topology τ on a Riesz space X is said to be locally solid [48] if τ has a base at zero consisting of solid sets. A locally solid Riesz space (X,τ) is a Riesz space equipped with a locally solid topology τ.

Recall [49] that a topological space is first countable if each point has a countable (decreasing) local base.

The purpose of this paper is to give certain characterizations of lacunary statistically convergent double sequences in locally solid Riesz spaces and obtain extensions of a decomposition theorem and some inclusion results related to the notions statistically convergence and lacunary statistically convergence in locally solid Riesz spaces.

Throughout the paper, the symbol Nsol will denote any base at zero consisting of solid sets and satisfying the conditions (1), (2), and (3) in a locally solid topology.

4. Double Lacunary Statistical Convergence in Locally Solid Riesz Spaces

Throughout the paper X will denote the Hausdorff locally solid Riesz space which is first countable.

The idea of lacunary statistical convergence for single sequences in locally solid Riesz spaces has been recently studied by Mohiuddine and Alghamdi [50] as follows.

Definition 5 (see [<xref ref-type="bibr" rid="B50">50</xref>]).

Let (X,τ) be a locally solid Riesz space. A sequence (xk) of points in X is said to be Sθ(τ)-convergent to an element x0 of X if for each τ-neighborhood V of zero,
(10)δθ({k∈ℕ:xk-x0∉V})=0;
that is,
(11)limr1hr|{k∈Ir:xk-x0∉V}|=0.
In this case, we write Sθ(τ)-limk→∞xk=x0or(xk)→Sθ(τ)x0.

Albayrak and Pehlivan [7] introduced the notion of statistical convergence in locally solid Riesz spaces. Afterward, Mohiuddine et al. [22] defined and studied the concept of statistical convergence in this setup as follows.

Definition 6 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Let (X,τ) be a locally solid Riesz space. Then, a double sequence x=(xjk) in X is said to be statistically τ-convergent to the number x0∈X if for every τ-neighborhood V of zero,
(12)P-limm,n→∞1mn|{(j,k),j≤m,k≤n:xjk-x0∉V}|=0.
In this case we write S(τ)-limx=ξ or xjk→𝒮(τ)x0.

Now we recall the definition of lacunary statistical convergence of double sequences in the framework of locally solid Riesz spaces as follows.

Definition 7.

Let (X,τ) be a locally solid Riesz space. A double sequence (xk,l) of points in X is said to be double lacunary statisticalτ-convergent or Sθ¯(τ)-convergent to an element x0 of X if for each τ-neighborhood V of zero,
(13)δθ¯({(k,l)∈ℕ×ℕ:xk,l-x0∉V})=0;
that is
(14)P-limr,s1hr,s|{(k,l)∈Ir,s:xk,l-x0∉V}|=0.
In this case, we write Sθ¯(τ)-limk,l→∞xk,l=x0 or (xk,l)→Sθ¯(τ)x0.

Now we prove our results.

Theorem 8.

Let (X,τ) be a locally solid Riesz space. If a double sequence (xk,l) of points in X is Sθ¯(τ)-convergent to x0 in X, then there are double sequences (yk,l) and (zk,l) such that Sθ¯(τ)-limk,l→∞yk,l=x0 and xk,l=yk,l+zk,l, for all (k,l)∈ℕ×ℕ and δθ¯({(k,l)∈ℕ×ℕ:xk,l≠yk,l})=0 and (zk,l) is a Sθ¯(τ)-null sequence.

Proof.

Let {Vi} be a nested base of τ-neighborhoods of zero. Take n0=0 and choose an increasing sequence (ni) of positive integers such that
(15)δθ¯({(k,l)∈ℕ×ℕ:xk,l-x0∉Vi})<1ifork,l>ni.
Let us define the sequences (yk,l) and (zk,l) as follows:
(16)yk,l=xk,l,zk,l=0,if0<k,l≤n1
and suppose ni<ni+1, for i≥1,
(17)yk,l=xk,l,zk,l=0,ifxk,l-x0∈Vi,yk,l=x0,zk,l=xk,l-x0,ifxk,l-x0∉Vi.

To show that, (i) P-limk,l→∞yk,l=x0 and (ii) (zk,l) is a Sθ¯(τ)-null sequence.

Let V be an arbitrary τ-neighborhood of zero. Since X is first countable, we may choose a positive integer i such that Vi⊆V. Then yk,l-x0=xk,l-x0∈Vi, for k,l>ni.

If xk,l-x0∉Vi, then yk,l-x0=x0-x0=0∈V. Hence P-limk,l→∞yk,l=x0.

It is enough to show that δθ¯({(k,l)∈ℕ×ℕ:zk,l≠0})=0. For any τ-neighborhood V of zero, we have
(18)δθ¯({(k,l)∈ℕ×ℕ:zk,l∉V})≤δθ¯({(k,l)∈ℕ×ℕ:zk,l≠0}).

If np<k,l≤np+1, then
(19){(k,l)∈ℕ×ℕ:zk,l≠0}⊆{(k,l)∈ℕ×ℕ:xk,l-x0∉Vp}.
If p>i and np<k,l≤np+1, then
(20)δθ¯({(k,l)∈ℕ×ℕ:zk,l≠0})≤δθ¯({(k,l)∈ℕ×ℕ:xk,l-x0∉Vp})<1p<1i<ε.
This implies that δθ¯({(k,l)∈ℕ×ℕ:zk,l≠0})=0. Hence (zk,l) is a Sθ¯(τ)-null sequence.
Theorem 9.

Let (X,τ) be a locally solid Riesz space and let x=(xk,l) be a double sequence of points in X. If there is a Sθ¯(τ)-convergent sequence y=(yk,l) in X such that δθ¯({(k,l)∈ℕ×ℕ:yk,l≠xk,l∉V})=0, then x is also Sθ¯(τ)-convergent.

Proof.

Suppose that δθ¯({(k,l)∈ℕ×ℕ:yk,l≠xk,l∉V})=0 and Sθ¯(τ)-limk,lyk,l=x0. Then for an arbitrary τ-neighborhood V of zero, we have
(21)δθ¯({(k,l)∈ℕ×ℕ:yk,l-x0∉V})=0.
Now,
(22){(k,l)∈ℕ×ℕ:xk,l-x0∉V}⊆{(k,l)∈ℕ×ℕ:yk,l≠xk,l∉V}∪{(k,l)∈ℕ×ℕ:yk,l-x0∉V}⟹δθ¯({(k,l)∈ℕ×ℕ:xk,l-x0∉V})≤δθ¯({(k,l)∈ℕ×ℕ:yk,l≠xk,l∉V})+δθ¯({(k,l)∈ℕ×ℕ:yk,l-x0∉V}).
Therefore, we have
(23)δθ¯({(k,l)∈ℕ×ℕ:xk,l-x0∉V})=0.

This completes the proof of the theorem.

5. Some Inclusions Relations in Locally Solid Riesz Spaces

Here, we prove some inclusion type results. We begin with the following interesting result.

Theorem 10.

Let (X,τ) be a locally solid Riesz space and let x=(xk,l) be a double sequence of points in X. For any double lacunary sequence θ¯={(kr,ls)},S(τ)⊆Sθ¯(τ) if and only if P-
lim
infr,sqr,s>1.

Proof.

Suppose first that P-liminfr,sqr,s>1, and P-liminfr,sqr,s=a (say). Write b=(a-1)/2. Then there exists an integer n0,m0∈ℕ such that qr,s≥1+b for r≥n0;s≥m0. Hence for r≥n0;s≥m0,
(24)hr,skrls=1-kr-1ls-1krls=1-1qr,s≥1-11+b=b1+b.
Suppose that S(τ)-limk,lxk,l=x0. We prove that Sθ¯(τ)-limk,lxk,l=x0. Let V be an arbitrary τ-neighborhood of zero. Then for all r≥n0;s≥m0, we have
(25)1kr,s|{k≤kr,l≤ls:xk,l-x0∉V}|≥1kr,s|{(k,l)∈Ir,s:xk,l-x0∉V}|=hr,skr,s1hr,s|{(k,l)∈Ir,s:xk,l-x0∉V}|≥b1+b1hr,s|{(k,l)∈Ir,s:xk,l-x0∉V}|.

Since (xk,l)→S(τ)x0. Therefore this inequality implies that (xk,l)→Sθ¯(τ)x0. Hence S(τ)⊆Sθ¯(τ).

Next, we suppose that P-liminfr,sqr,s=1. We can select a subsequence {(kri,lsj)} of the double lacunary sequence θ¯ such that
(26)krilsjkri-1lsj-1<1+1ij,kri-1lsj-1kri-1lsj-1>ij,
where ri>ri-1+2, sj>sj-1+2. Take a(≠0)∈X. Now we define a sequence (xk,l) by
(27)xk,l={a,if(k,l)∈Iri,sjforsomei,j=1,2,3,…0,otherwise.
Then S(τ)-limk,lxk,l=0. To see this, let V be an arbitrary τ-neighborhood of zero. We choose W∈Nsol such that W⊆V and a∉W. On the other hand, for each m,n we can find a positive number (im,jn) such that krim<m≤krim+1, lsjn<n≤lsjn+1. Then
(28)1mn|{k≤m,l≤n:xk,l∉V}|≤1krimlsjn|{k≤m,l≤n:xk,l∉W}|≤1krimlsjn{|{k≤krim,l≤lsjn:xk,l∉W}|+|{krim<k≤m,lsjn<l≤n:xk,l∉W}|}≤1krim,lsjn|{k≤krim,l≤lsjn:xk,l∉W}|+1krim,lsjn(krim+1-krim)(lsin+1-lsin)<1imjn+1+1imjn-1<1(im+1)(jn+1)+1imjnforeachm,n.
Therefore S(τ)-limk,lxk,l=0. Now let us see that (xk,l)∉Sθ¯(τ). Let V be a τ-neighborhood of zero such that a∉V. Thus
(29)P-limi,j→∞1hri,sj|{kri-1<k≤kri,lsj-1<l≤lsj:xk,l∉V}|=P-limi,j→∞1hri,sj(kri,sj-kri-1,sj-1)=P-limi,j→∞1hri,sjhri,sj=1
and for r≠ri,s≠sj,i,j=1,2,3,…,
(30)P-limr,s→∞1hr,s|{kr-1<k≤kr,ls-1<l≤ls:xk,l-a∉V}|=1.
Hence neither a nor 0 can be double lacunary statistical limit of (xk,l). No other point of X can be double lacunary statistical limit of the sequence (xk,l) as well. Thus (xk,l)∉Sθ¯(τ). This completes the proof of the theorem.

Theorem 11.

Let (X,τ) be a locally solid Riesz space and let x=(xk,l) be sequence in X. For any double lacunary sequence θ¯={(kr,ls)},Sθ¯(τ)⊆S(τ) if and only if P-
lim
supr,sqr,s<∞.

Proof.

Suppose that P-limsupr,sqr,s<∞. Then there exists an H>0 such that qr,s<H for all r,s. Let Sθ¯(τ)-limk,lxk,l=x0. Let V be an arbitrary τ-neighborhood of zero. Let ε>0. We write
(31)Mr,s={(k,l)∈Ir,s:xk,l-x0∉V}.
By the definition of double lacunary statistical convergence, there are positive numbers r0, s0 such that
(32)Mr,shr,s<ε2H∀r>r0,s>s0.
Let M=max{Mr,s:1≤r≤r0,1≤s≤s0} and let m,n be two integers satisfying kr-1<m≤kr;ls-1<n≤ls; then we can write
(33)1mn|{k≤m,l≤n:xk,l-x0∉V}|≤r0s0Mkr-1ls-1+ε12Hqr,s.
Since P-limr,s→∞kr,s=∞, there exist positive integers r1≥r0,s1≥s0 such that
(34)1kr-1ls-1<ε2r0s0Mforr>r1,s>s1.
Hence for r>r1,s>s1(35)1mn|{k≤m,l≤n:xk,l-x0∉V}|<ε2+ε2=ε.
It follows that S(τ)-limk,lxk,l=x0.

Next we suppose that P-limsupr,sqr,s=∞. Take an element a≠0∈X. Let {(kri,lsj)} be a subsequence of the double lacunary sequence θ¯={(kr,ls)} such that qri,sj>ij,kri>i+3,lsj>j+3. Define a sequence (xk,l) by
(36)xk,l={a,ifkri-1<k≤2kri-1;lsj-1<l≤2lsj-1forsomei,j=1,2,3,…0,otherwise.
Let V be a τ-neighborhood of zero such that a∉V. Then for i,j>1(37)1hri,sj|{k≤kri,l≤lsj:xk,l∉V}|<kri,sj-1hri,sj=kri,sj-1kri,sj-kri,sj-1<1ij-1.
Hence (xk,l)∈Sθ¯(τ). But (xk,l)∉S(τ), because
(38)12kri,sj-1|{k≤2kri-1,l≤2lsj-1:xk,l∉V}|=12kri,sj-1[kr1,s1-1+kr2,s2-1+⋯+kri,sj-1]>12.
This completes the proof of the theorem.

Corollary 12.

Let (X,τ) be a locally solid Riesz space and let x=(xk,l) be a double sequence in X. For any double lacunary sequence θ¯={(kr,ls)},Sθ¯(τ)=S(τ) if and only if 1<P-
lim
infr,sqr,s≤P-
lim
supr,sqr,s<∞.

Theorem 13.

Let (X,τ) be a locally solid Riesz space, let x=(xk,l) be a double sequence in X. For any double lacunary sequence θ¯={(kr,ls)}, if x=(xk,l)∈Sθ¯(τ)∩S(τ), then S(τ)-limk,l→∞xk,l=Sθ¯(τ)-limk,l→∞xk,l.

Proof.

Let x=(xk,l)∈Sθ¯(τ)∩S(τ) and S(τ)-limk,l→∞xk,l=x0, and Sθ¯(τ)-limk,l→∞xk,l=y0. Suppose that x0≠y0. Since X is a Hausdorff, then there exists a τ-neighborhood V of zero value such that x0-y0∉V. We choose W∈Nsol such that W+W⊆V. Then, we have
(39)1kmln|{k≤km,l≤ln:x0-y0∉V}|≤1kmln|{k≤km,l≤ln:xk,l-x0∉W}|+1kmln|{k≤km,l≤ln:y0-xk,l∉W}|.

It follows from this inequality that
(40)1≤1kmln|{k≤km,l≤ln:xk,l-x0∉W}|+1kmln|{k≤km,l≤ln:y0-xk,l∉W}|.
We write
(41)1kmln|{k≤km,l≤ln:y0-xk,l∉W}|=1km,ln|{(k,l)∈⋃r,s=1m,nIr,s:y0-xk,l∉W}|=1kmln∑r,s=1m,n|{(k,l)∈Ir,s:y0-xk,l∉W}|=(∑r,s=1m,nhr,s)-1(∑r,s=1m,nhr,s·Tr,s),
where
(42)Tr,s=1hr,s|{(k,l)∈Ir,s:y0-xk,l∉W}|.
Since Sθ¯(τ)-limk,l→∞xk,l=y0, we have P-limr,s→∞Tr,s=0. Therefore the regular weighted mean transform of (Tr,s) also tends to 0; that is,
(43)P-limm,n→∞1kmln|{k≤km,l≤ln:y0-xk,l∉W}|=0.
Also since S(τ)-limk,l→∞xk,l=x0, we have
(44)P-limm,n→∞1kmln|{k≤km,l≤ln:xk,l-x0∉W}|=0.
From (39), (43), and (44), we have
(45)1kmln|{k≤km,l≤ln:x0-y0∉V}|=0.
This contradiction completes the proof of the theorem.

6. Double Statistical Lacunary Summable in Locally Solid Riesz Spaces

In this section, we introduce some new concepts by using the notions of statistical lacunary summable for double sequences.

Definition 14.

Let (X,τ) be a locally solid Riesz space. A sequence x=(xk,l) is said to be double lacunary summable (or shortly, θ-summable) in (X,τ) or simply θτ-summable to an element x0∈X if for each τ-neighborhood V of zero value that tr,s(x)-x0∈V, where tr,s(x)=(1/hr,s)∑(k,l)∈Ir,sxk,l. In this case, we write θτ-limx=x0.

Definition 15.

Let (X,τ) be a locally solid Riesz space. A sequence (xk,l) of points in X is said to be double statistical lacunaryτ-summable or simply Sδθ¯(τ)-summable to an element x0 of X if for each τ-neighborhood V of zero value, the set K(θ¯)={(r,s)∈ℕ×ℕ:tr,s(x)-x0∉V} has double natural density zero; that is, δ2(K(θ¯))=0

That is
(46)P-limmn1mn|{r≤m,s≤n:tr,s(x)-x0∉V}|=0.
In this case, we write Sδθ¯(τ)-limxk,l=x0or(xk,l)→Sδθ¯(τ)x0.

Theorem 16.

Let (X,τ) be a locally solid Riesz space. A double sequence x=(xk,l) in X is Sδθ¯(τ)-summable to x0 if and only if there exists a set K={(r,s)}⊆ℕ×ℕ, r,s=1,2,…, such that δ2(K)=1 and θτ-lim(r,s)∈Kxr,s=x0.

Proof.

Let V be an arbitrary τ-neighborhood of zero. Suppose that θτ-lim(r,s)∈Kxr,s=x0; there exists a set K={(r,s)}⊆ℕ×ℕ, r,s=1,2,…, with δ2(K)=1 and N=N(V), M=M(V) such that (tr,s(x)-x0)∈V for r>N and s>M. Write KV={(r,s)∈ℕ×ℕ:tr,s(x)-x0∉V} and K1={(rN+1,sM+1),(rN+2,sM+2),…}. Then δ2(K1)=1 and KV⊆ℕ×ℕ-K1 which implies that δ2(KV)=0. Hence x=(xk,l) is Sδθ¯(τ)-summable to x0.

Conversely suppose that x=(xk,l) is Sδθ¯(τ)-convergent to x0. Fix a countable local base V1⊃V2⊃⋯ at x0. For each i∈ℕ, put
(47)Ki={(r,s)∈ℕ×ℕ:tr,s(x)-x0∉Vi}.
By hypothesis δ2(Ki)=0 for each i. Since the ideal ℐ of all subsets of ℕ×ℕ having double density zero is a P-ideal (see e.g., [51]), then there exists a sequence of sets (Ji)i such that the symmetric difference KiΔJi is a finite set for any i∈ℕ and J:=∪i=1∞Ji∈ℐ.

Let K=ℕ×ℕ∖J, then δ(K)=1. In order to prove the theorem, it is enough to check that lim(r,s)∈Ktr,s(x)=x0.

Let i∈ℕ. Since KiΔJi is a finite, there is (ri,si)∈ℕ×ℕ, without loss of generality with (ri,si)∈K, ri,si>i, such that
(48)(ℕ×ℕ∖Ji)∩{(r,s)∈ℕ×ℕ:r≥ri,s≥si}=(ℕ×ℕ∖Ki)∩{(r,s)∈ℕ×ℕ:r≥ri,s≥si}.
If (r,s)∈K and r≥ri,s≥si then (r,s)∉Ji, and by (48) (r,s)∉Ki. Thus tr,s(x)-x0∈Vi. So we have proved that for all i∈ℕ there is (ri,si)∈K,ri,si>i, with tr,s(x)-x0∈Vi for every r≥ri,s≥si: without loss of generality, we can suppose ri+1>ri and si+1>si for every i∈ℕ. The assertion follows taking into account that the Vi′s form a countable local base at x0.

Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

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