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Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces

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Abstract

The notion of statistical convergence was defined by Fast (Colloq. Math. 2:241-244, 1951) and over the years was further studied by many authors in different setups. In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and S (τ)-convergence through de la Vallée-Poussin mean in a locally solid Riesz space.

MSC:40A35, 40G15, 46A40.

1 Introduction and preliminaries

Since 1951, when Steinhaus [1] and Fast [2] defined statistical convergence for sequences of real numbers, several generalizations and applications of this notion have been investigated. For more detail and related concepts, we refer to [329] and references therein. Quite recently, Di Maio and Kǒcinac [30] studied this notion in topological and uniform spaces and Albayrak and Pehlivan [31], and Mohiuddine and Alghamdi [32] for real and lacunary sequences, respectively, in locally solid Riesz spaces. Afterward, the idea was extended to double sequences by Mohiuddine et al.[33] in the framework of locally solid Riesz spaces.

Let K be a subset of , the set of natural numbers. Then the asymptotic density of K denoted by δ(K) is defined as

δ(K)= lim n 1 n | { k n : k K } | ,

where the vertical bars denote the cardinality of the enclosed set.

The number sequence x=( x j ) is said to be statistically convergent to the number if for each ϵ>0,

lim n 1 n | { j n : | x j | ϵ } | =0.

In this case, we write st-lim x j =.

Remark 1.1 It is well known that every statistically convergent sequence is convergent, but the converse is not true. For example, suppose that the sequence x=( x n ) is defined as

x=( x n )={ n if  n  is a square , 0 otherwise .

It is clear that the sequence x=( x n ) is statistically convergent to 0, but it is not convergent.

Now we recall some definitions related to the notion of a locally solid Riesz space. Let X be a real vector space and ≤ be a partial order on this space. Then X is said to be an ordered vector space if it satisfies the following properties:

  1. (i)

    If x,yX and yx, then y+zx+z for each zX.

  2. (ii)

    If x,yX and yx, then λyλx for each λ0.

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) [34].

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

A subset S of a Riesz space X is said to be solid if yS and |x||y| imply xS.

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

Every linear topology τ on a vector space X has a base N for the neighborhoods of θ satisfying the following properties:

(C1) Each YN is a balanced set, that is, λxY holds for all xY and every λR with |λ|1.

(C2) Each YN is an absorbing set, that is, for every xX, there exists λ>0 such that λxY.

(C3) For each YN, there exists some EN with E+EY.

A linear topology τ on a Riesz space X is said to be locally solid (cf.[35, 36]) 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.

In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and S (τ)-convergence through de la Vallée-Poussin mean in a locally solid Riesz space.

2 Generalized statistical τ-convergence

Throughout the text, we write N sol for any base at zero consisting of solid sets and satisfying the conditions (C1), (C2) and (C3) in a locally solid topology. The following idea of λ-statistical convergence was introduced in [37] and further studied in [3840].

Let λ=( λ n ) be a non-decreasing sequence of positive numbers tending to ∞ such that

λ n + 1 λ n +1, λ 1 =0.

The generalized de la Vallée-Poussin mean is defined by

t n (x)=: 1 λ n j I n x j ,

where I n =[n λ n +1,n].

A sequence x=( x j ) is said to be (V,λ)-summable to a number if

t n (x)as n.

A sequence x=( x j ) is said to be strongly(V,λ)-summable to a number if

1 λ n j I n | x j |0as n.

We denote it by x j [V,λ] as j.

Let KN be a set of positive integers, then

δ λ (K)= lim n 1 λ n | { n λ n + 1 j n : j K } |

is said to be the λ-density of K.

In case λ n =n, the λ-density reduces to the natural density.

The number sequence x=( x j ) is said to be λ-statistically convergent to the number if for each ϵ>0, δ λ ( K ϵ )=0, where K ϵ ={jN:| x j |>ϵ}, i.e.,

lim n 1 λ n | { j I n : | x j | > ϵ } | =0.

In this case, we write st λ - lim j x j = and we denote the set of all λ-statistically convergent sequences by S λ . This notion was extended to double sequences in [41, 42].

Remark 2.1 As in Remark 1.1, we observe that if a sequence is (V,λ)-summable to a number , then it is also λ-statistically convergent to the same number , but the converse need not be true. For example, let the sequence z=( z k ) be defined by

z k ={ k if  n [ λ n ] + 1 k n , 0 otherwise ,

where [a] denotes the integer part of aR. Then x is λ-statistically convergent to 0 but not (V,λ)-summable.

Definition 2.1 Let (X,τ) be a locally solid Riesz space. Then a sequence x=( x j ) in X is said to be generalized statistically τ-convergent (or S λ (τ)-convergent) to the number ξX if for every τ-neighborhood U of zero,

lim n 1 λ n | { j I n : x j ξ U } | =0.

In this case, we write S λ (τ)-limx=ξ or x j S λ ( τ ) ξ.

Definition 2.2 Let (X,τ) be a locally solid Riesz space. We say that a sequence x=( x j ) in X is generalized statistically τ-bounded if for every τ-neighborhood U of zero, there exists some λ>0 such that the set

{jN:λ x j U}

has λ-density zero.

Theorem 2.1 Let(X,τ)be a Hausdorff locally solid Riesz space andx=( x j )andy=( y k )be two sequences in X. Then the following hold:

  1. (i)

    If S λ (τ)- lim j x j = ξ 1 and S λ (τ)- lim j x j = ξ 2 , then ξ 1 = ξ 2 .

  2. (ii)

    If S λ (τ)- lim j x j =ξ, then S λ (τ)- lim j α x j =αξ, αR.

  3. (iii)

    If S λ (τ)- lim j x j =ξ and S λ (τ)- lim j y j =η, then S λ (τ)- lim j ( x j + y j )=ξ+η.

Proof (i) Suppose that S λ (τ)- lim j x j = ξ 1 and S λ (τ)- lim j x j = ξ 2 . Let U be any τ-neighborhood of zero. Then there exists Y N sol such that YU. Choose any E N sol such that E+EY. We define the following sets:

K 1 = { j N : x j ξ 1 E } , K 2 = { j N : x j ξ 2 E } .

Since S λ (τ)- lim j x j = ξ 1 and S λ (τ)- lim j x j = ξ 2 , we have δ λ ( K 1 )= δ λ ( K 2 )=1. Thus δ( K 1 K 2 )=1 and, in particular, K 1 K 2 . Now, let j K 1 K 2 . Then

ξ 1 ξ 2 = ξ 1 x j + x j ξ 2 E+EYU.

Hence, for every τ-neighborhood U of zero, we have ξ 1 ξ 2 U. Since (X,τ) is Hausdorff, the intersection of all τ-neighborhoods U of zero is the singleton set {θ}. Thus, we get ξ 1 ξ 2 =θ, i.e., ξ 1 = ξ 2 .

  1. (ii)

    Let U be an arbitrary τ-neighborhood of zero and S λ (τ)- lim j x j =ξ. Then there exists Y N sol such that YU and also

    lim n 1 λ n | { j I n : x j ξ Y } | =1.

Since Y is balanced, x j ξY implies α( x j ξ)Y for every αR with |α|1. Hence, for every nN, we get

{ j I n : x j ξ Y } { j I n : α x j α ξ Y } { j I n : α x j α ξ U } .

Thus, we obtain

lim n 1 λ n | { j I n : α x j α ξ U } | =1

for each τ-neighborhood U of zero. Now let |α|>1 and [|α|] be the smallest integer greater than or equal to |α|. There exists E N sol such that [|α|]EY. Since S λ (τ)- lim j x j =ξ, the set

K={jN: x j ξE}

has λ-density zero. Therefore, for all nN and jK I n , we have

|αξα x j |=|α||ξ x j | [ | α | ] |ξ x j | [ | α | ] EYU.

Since the set Y is solid, we have αξα x j Y. This implies that αξα x j U. Thus,

lim n 1 λ n | { j I n : α x j α ξ U } | =1

for each τ-neighborhood U of zero. Hence S λ (τ)- lim j α x j =αξ.

  1. (iii)

    Let U be an arbitrary τ-neighborhood of zero. Then there exists Y N sol such that YU. Choose E in N sol such that E+EY. Since S λ (τ)- lim j x j =ξ and S λ (τ)- lim j y j =η, we have δ λ ( H 1 )=1= δ λ ( H 2 ), where

    H 1 = { j N : x j ξ E } , H 2 = { j N : y j η E } .

Let H= H 1 H 2 . Hence, we have δ λ (H)=1. For all nN and jH I n , we get

( x j + y j )(ξ+η)=( x j ξ)+( y j η)E+EYU.

Therefore,

lim n 1 λ n | { j I n : ( x j + y j ) ( ξ + η ) U } | =1.

Since U is arbitrary, we have S λ (τ)- lim j ( x j + y j )=ξ+η. □

Theorem 2.2 Let(X,τ)be a locally solid Riesz space. If a sequencex=( x j )is generalized statistically τ-convergent, then it is generalized statistically τ-bounded.

Proof Suppose x=( x j ) is generalized statistically τ-convergent to the point ξX and let U be an arbitrary τ-neighborhood of zero. Then there exists Y N sol such that YU. Let us choose E N sol such that E+EY. Since S λ (τ)- lim j x j =ξ, the set

K={jN: x j ξE}

has λ-density zero. Since E is absorbing, there exists λ>0 such that λξE. Let α(0,min{1,λ}). Since E is solid and |αξ||λx|, we have αξE. Since E is balanced, x j ξE implies α( x j ξ)E. Then, for each nN and j(NK) I n , we have

α x j =α( x j ξ)+αξE+EYU.

Thus

lim n 1 λ n | { j I n : α x j U } | =0.

Hence, ( x j ) is generalized statistically τ-bounded. □

Theorem 2.3 Let(X,τ)be a locally solid Riesz space. If( x j ), ( y j )and( z j )are three sequences such that

  1. (i)

    x j y j z j for all jN,

  2. (ii)

    S λ (τ)- lim j x j =ξ= S λ (τ)- lim j z j ,

then S λ (τ)- lim j y j =ξ.

Proof Let U be an arbitrary τ-neighborhood of zero, there exists Y N sol such that YU. Choose E N sol such that E+EY. From condition (ii), we have δ λ (A)=1= δ λ (B), where

A = { j N : x j ξ E } , B = { j N : x j ξ E } .

Also, we get δ λ (AB)=1, and from (i) we have

x j ξ y j ξ z j ξ

for all jN. This implies that for all nN and jAB I n , we get

| y j ξ|| x j ξ|+| z j ξ|E+EY.

Since Y is solid, we have y j ξYU. Thus,

lim n 1 λ n | { j I n : y j ξ U } | =1

for each τ-neighborhood U of zero. Hence S λ (τ)- lim j y j =ξ. □

3 Generalized statistically τ-Cauchy and S λ (τ)-convergence

Definition 3.1 Let (X,τ) be a locally solid Riesz space. A sequence x=( x j ) in X is generalized statistically τ-Cauchy if for every τ-neighborhood U of zero there exists pN such that the set

{jN: x j x p U}

has λ-density zero.

Theorem 3.1 Let(X,τ)be a locally solid Riesz space. If a sequencex=( x j )is generalized statistically τ-convergent, then it is generalized statistically τ-Cauchy.

Proof Suppose that S λ (τ)- lim j x j =ξ. Let U be an arbitrary τ-neighborhood of zero, there exists Y N sol such that YU. Choose E N sol such that E+EY. By generalized statistical τ-convergence to ξ, there is pN with ξ x p E and

lim n 1 λ n | { j I n : x j ξ E } | =0.

Also, for all nN and j(NK) I n , where

K={jN: x j ξE},

we have

x j x p = x j ξ+ξ x p E+EYU

and δ λ (K)=0. Therefore the set

{jN: x j x p U}K I n

for all nN. For every τ-neighborhood U of zero there exists pN such that the set {jN: x j x p U} has λ-density zero. Hence ( x j ) is generalized statistically τ-Cauchy. □

Now we define another type of convergence in locally solid Riesz spaces.

Definition 3.2 A sequence ( x j ) in a locally solid Riesz space (X,τ) is said to be S λ (τ)-convergent to ξX if there exists an index set K={ j n }N, n=1,2, , with δ λ (K)=1 such that lim n x j n =ξ. In this case, we write ξ= S λ (τ)-limx.

Theorem 3.2 A sequencex=( x j )in a locally solid Riesz space(X,τ)is generalized statistically τ-convergent to a number ξ if it is S λ (τ)-convergent to ξ.

Proof Let U be an arbitrary τ-neighborhood of ξ. Since x=( x j ) is S λ (τ)-convergent to ξ, there is an index set K={ j n }N, n=1,2, , with δ λ (K)=1 and j 0 = j 0 (U), such that j j 0 and jK imply x j ξU. Then

K U ={jN: x j ξU}N{ j N + 1 , j N + 2 ,}.

Therefore δ λ ( K U )=0. Hence x is generalized statistically τ-convergent to ξ. □

Note that the converse holds for a first countable space.

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

Theorem 3.3 Let(X,τ)be a first countable locally solid Riesz space. If a sequencex=( x j )is generalized statistically τ-convergent to a number ξ, then it is S λ (τ)-convergent to ξ.

Proof Let x be generalized statistically τ-convergent to a number ξ. Fix a countable local base U 1 U 2 U 3 at ξ. For each iN, put

K i ={jN: x j ξ U i }.

By hypothesis, δ λ ( K i )=0 for each i. Since the ideal of all subsets of having λ-density zero is a P-ideal (see, for instance, [43]), then there exists a sequence of sets ( J i ) i such that the symmetric difference K i Δ J i is a finite set for any iN and J:= i = 1 J i I.

Let K=NJ, then δ λ (K)=1. In order to prove the theorem, it is enough to check that lim j K x j =ξ.

Let iN. Since K i Δ J i is finite, there is j i N, without loss of generality, with j i K, j i >i, such that

(N J i ){jN:j j i }=(N K i ){jN:j j i }.
(1)

If jK and j j i , then j J i , and by (1), j K i . Thus x j ξ U i . So, we have proved that for all iN, there is j i K, j i >i, with x j ξ U i for every j j i : without loss of generality, we can suppose j i + 1 > j i for every iN. The assertion follows taking into account that the U i ’s form a countable local base at ξ. □

4 Conclusion

Recently, statistical convergence has been established as a better option than ordinary convergence. It is found very interesting that some results on sequences, series and summability can be proved by replacing the ordinary convergence by statistical convergence; and further, through some examples, where some efforts are required, we can show that the results for statistical convergence happen to be stronger than those proved for ordinary convergence (e.g., [4449]). This notion has also been defined and studied in different setups. In this paper, we have studied this notion through de la Vallée-Poussin mean in a locally solid Riesz space to deal with the convergence problems in a broader sense.

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Acknowledgements

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

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Correspondence to Syed Abdul Mohiuddine.

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The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Keywords

  • statistical convergence
  • statistical Cauchy
  • de la Vallée-Poussin mean
  • locally solid Riesz space