Skip to content

Advertisement

  • Research
  • Open Access

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

  • 1Email author,
  • 1 and
  • 2
Advances in Difference Equations20132013:66

https://doi.org/10.1186/1687-1847-2013-66

Received: 16 November 2012

Accepted: 1 March 2013

Published: 21 March 2013

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.

Keywords

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

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 , y X and y x , then y + z x + z for each z X .

     
  2. (ii)

    If x , y X and y x , then λ y λ x for each λ 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) [34].

For an element x of a Riesz space X, the positive part of x is defined by x + = x θ = 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 y S and | x | | y | imply x S .

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 × X X defined by f ( x , y ) = x + y is continuous on X × X , and the continuity of scalar multiplication means that the function f : R × X X defined by f ( λ , x ) = λ x 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 Y N is a balanced set, that is, λ x Y holds for all x Y and every λ R with | λ | 1 .

(C2) Each Y N is an absorbing set, that is, for every x X , there exists λ > 0 such that λ x Y .

(C3) 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 (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 | 0 as  n .

We denote it by x j [ V , λ ] as j .

Let K N 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 = { j N : | 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 a R . 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 ξ X 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 ) - lim x = 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
{ j N : λ x j U }

has λ-density zero.

Theorem 2.1 Let ( X , τ ) be a Hausdorff locally solid Riesz space and x = ( x j ) and y = ( 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 = α ξ , α R .

     
  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 Y U . Choose any E N sol such that E + E Y . 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 + E Y U .
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 Y U and also
    lim n 1 λ n | { j I n : x j ξ Y } | = 1 .
     
Since Y is balanced, x j ξ Y implies α ( x j ξ ) Y for every α R with | α | 1 . Hence, for every n N , 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 | ] E Y . Since S λ ( τ ) - lim j x j = ξ , the set
K = { j N : x j ξ E }
has λ-density zero. Therefore, for all n N and j K I n , we have
| α ξ α x j | = | α | | ξ x j | [ | α | ] | ξ x j | [ | α | ] E Y U .
Since the set Y is solid, we have α ξ α x j Y . This implies that α ξ α 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 = α ξ .
  1. (iii)
    Let U be an arbitrary τ-neighborhood of zero. Then there exists Y N sol such that Y U . Choose E in N sol such that E + E Y . 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 n N and j H I n , we get
( x j + y j ) ( ξ + η ) = ( x j ξ ) + ( y j η ) E + E Y U .
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 sequence x = ( 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 ξ X and let U be an arbitrary τ-neighborhood of zero. Then there exists Y N sol such that Y U . Let us choose E N sol such that E + E Y . Since S λ ( τ ) - lim j x j = ξ , the set
K = { j N : x j ξ E }
has λ-density zero. Since E is absorbing, there exists λ > 0 such that λ ξ E . Let α ( 0 , min { 1 , λ } ) . Since E is solid and | α ξ | | λ x | , we have α ξ E . Since E is balanced, x j ξ E implies α ( x j ξ ) E . Then, for each n N and j ( N K ) I n , we have
α x j = α ( x j ξ ) + α ξ E + E Y U .
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 j N ,

     
  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 Y U . Choose E N sol such that E + E Y . 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 ( A B ) = 1 , and from (i) we have
x j ξ y j ξ z j ξ
for all j N . This implies that for all n N and j A B I n , we get
| y j ξ | | x j ξ | + | z j ξ | E + E Y .
Since Y is solid, we have y j ξ Y U . 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 p N such that the set
{ j N : x j x p U }

has λ-density zero.

Theorem 3.1 Let ( X , τ ) be a locally solid Riesz space. If a sequence x = ( 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 Y U . Choose E N sol such that E + E Y . By generalized statistical τ-convergence to ξ, there is p N with ξ x p E and
lim n 1 λ n | { j I n : x j ξ E } | = 0 .
Also, for all n N and j ( N K ) I n , where
K = { j N : x j ξ E } ,
we have
x j x p = x j ξ + ξ x p E + E Y U
and δ λ ( K ) = 0 . Therefore the set
{ j N : x j x p U } K I n

for all n N . For every τ-neighborhood U of zero there exists p N such that the set { j N : 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 ξ X 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 ) - lim x .

Theorem 3.2 A sequence x = ( 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 j K imply x j ξ U . Then
K U = { j N : x j ξ U } 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 sequence x = ( 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 i N , put
K i = { j N : 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 i N and J : = i = 1 J i I .

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

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

If j K 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 i N , 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 i N . 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.

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
(2)
Department of Mathematics, Aligarh Muslim University, Aligarh, India

References

  1. Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 2: 73-74.MathSciNetGoogle Scholar
  2. Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241-244.MathSciNetGoogle Scholar
  3. Çakalli H: Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math. 1995, 26(2):113-119.MathSciNetGoogle Scholar
  4. Çakalli H: On statistical convergence in topological groups. Pure Appl. Math. Sci. 1996, 43: 27-31.MathSciNetGoogle Scholar
  5. Çakalli H, Khan MK: Summability in topological spaces. Appl. Math. Lett. 2011, 24: 348-352. 10.1016/j.aml.2010.10.021MathSciNetView ArticleGoogle Scholar
  6. Çakalli H, Savaş E: Statistical convergence of double sequence in topological groups. J. Comput. Anal. Appl. 2010, 12(2):421-426.MathSciNetGoogle Scholar
  7. Edely OHH, Mursaleen M: On statistical A -summability. Math. Comput. Model. 2009, 49: 672-680. 10.1016/j.mcm.2008.05.053MathSciNetView ArticleGoogle Scholar
  8. Fridy JA: On statistical convergence. Analysis 1985, 5: 301-313.MathSciNetView ArticleGoogle Scholar
  9. Karakuş S, Demirci K: Statistical convergence of double sequences on probabilistic normed spaces. Int. J. Math. Math. Sci. 2007., 2007: Article ID 14737Google Scholar
  10. Karakuş S, Demirci K, Duman O: Statistical convergence on intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2008, 35: 763-769. 10.1016/j.chaos.2006.05.046MathSciNetView ArticleGoogle Scholar
  11. Maddox IJ: Statistical convergence in a locally convex space. Math. Proc. Camb. Philos. Soc. 1988, 104: 141-145. 10.1017/S0305004100065312MathSciNetView ArticleGoogle Scholar
  12. Mohiuddine SA, Aiyub M: Lacunary statistical convergence in random 2-normed spaces. Appl. Math. Inf. Sci. 2012, 6(3):581-585.MathSciNetGoogle Scholar
  13. Mohuiddine SA, Alotaibi A, Alsulami SM: Ideal convergence of double sequences in random 2-normed spaces. Adv. Differ. Equ. 2012., 2012: Article ID 149Google Scholar
  14. Mohiuddine SA, Danish Lohani QM: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Solitons Fractals 2009, 42: 1731-1737. 10.1016/j.chaos.2009.03.086MathSciNetView ArticleGoogle Scholar
  15. Mohiuddine SA, Savaş E: Lacunary statistically convergent double sequences in probabilistic normed spaces. Ann. Univ. Ferrara 2012, 58: 331-339. 10.1007/s11565-012-0157-5View ArticleGoogle Scholar
  16. Mohiuddine SA, Şevli H, Cancan M: Statistical convergence in fuzzy 2-normed space. J. Comput. Anal. Appl. 2010, 12(4):787-798.MathSciNetGoogle Scholar
  17. Mohiuddine SA, Şevli H, Cancan M: Statistical convergence of double sequences in fuzzy normed spaces. Filomat 2012, 26(4):673-681. 10.2298/FIL1204673MMathSciNetView ArticleGoogle Scholar
  18. Mursaleen M: On statistical convergence in random 2-normed spaces. Acta Sci. Math. 2010, 76: 101-109.MathSciNetGoogle Scholar
  19. Mursaleen M, Alotaibi A: On I -convergence in random 2-normed spaces. Math. Slovaca 2011, 61(6):933-940. 10.2478/s12175-011-0059-5MathSciNetView ArticleGoogle Scholar
  20. Mursaleen M, Edely OHH: Generalized statistical convergence. Inf. Sci. 2004, 162: 287-294. 10.1016/j.ins.2003.09.011MathSciNetView ArticleGoogle Scholar
  21. Mursaleen M, Edely OHH: Statistical convergence of double sequences. J. Math. Anal. Appl. 2003, 288: 223-231. 10.1016/j.jmaa.2003.08.004MathSciNetView ArticleGoogle Scholar
  22. Mursaleen M, Edely OHH: On the invariant mean and statistical convergence. Appl. Math. Lett. 2009, 22: 1700-1704. 10.1016/j.aml.2009.06.005MathSciNetView ArticleGoogle Scholar
  23. Mursaleen M, Mohiuddine SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 41: 2414-2421. 10.1016/j.chaos.2008.09.018MathSciNetView ArticleGoogle Scholar
  24. Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233: 142-149. 10.1016/j.cam.2009.07.005MathSciNetView ArticleGoogle Scholar
  25. Mursaleen M, Mohiuddine SA: On ideal convergence of double sequences in probabilistic normed spaces. Math. Rep. 2010, 12(64)(4):359-371.MathSciNetGoogle Scholar
  26. Mursaleen M, Mohiuddine SA: On ideal convergence in probabilistic normed spaces. Math. Slovaca 2012, 62: 49-62. 10.2478/s12175-011-0071-9MathSciNetView ArticleGoogle Scholar
  27. Mursaleen M, Mohiuddine SA, Edely OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput. Math. Appl. 2010, 59: 603-611. 10.1016/j.camwa.2009.11.002MathSciNetView ArticleGoogle Scholar
  28. Savaş E, Mohiuddine SA: λ ¯ -statistically convergent double sequences in probabilistic normed spaces. Math. Slovaca 2012, 62(1):99-108. 10.2478/s12175-011-0075-5MathSciNetGoogle Scholar
  29. Savaş E, Mursaleen M: On statistically convergent double sequences of fuzzy numbers. Inf. Sci. 2004, 162: 183-192. 10.1016/j.ins.2003.09.005View ArticleGoogle Scholar
  30. Di Maio G, Kočinac LDR: Statistical convergence in topology. Topol. Appl. 2008, 156: 28-45. 10.1016/j.topol.2008.01.015View ArticleGoogle Scholar
  31. Albayrak H, Pehlivan S: Statistical convergence and statistical continuity on locally solid Riesz spaces. Topol. Appl. 2012, 159: 1887-1893. 10.1016/j.topol.2011.04.026MathSciNetView ArticleGoogle Scholar
  32. Mohiuddine SA, Alghamdi MA: Statistical summability through a lacunary sequence in locally solid Riesz spaces. J. Inequal. Appl. 2012., 2012: Article ID 225Google Scholar
  33. Mohiuddine SA, Alotaibi A, Mursaleen M: Statistical convergence of double sequences in locally solid Riesz spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 719729Google Scholar
  34. Zaanen AC: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin; 1997.View ArticleGoogle Scholar
  35. Aliprantis CD, Burkinshaw O: Locally Solid Riesz Spaces with Applications to Economics. 2nd edition. Am. Math. Soc., Providence; 2003.View ArticleGoogle Scholar
  36. Roberts GT: Topologies in vector lattices. Math. Proc. Camb. Philos. Soc. 1952, 48: 533-546. 10.1017/S0305004100076295View ArticleGoogle Scholar
  37. Mursaleen M: λ -statistical convergence. Math. Slovaca 2000, 50: 111-115.MathSciNetGoogle Scholar
  38. Çolak R, Bektaş CA: λ -statistical convergence of order α . Acta Math. Sci., Ser. B 2011, 31(3):953-959.MathSciNetView ArticleGoogle Scholar
  39. Edely OHH, Mohiuddine SA, Noman AK: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 2010, 23: 1382-1387. 10.1016/j.aml.2010.07.004MathSciNetView ArticleGoogle Scholar
  40. de Malafosse B, Rakočević V: Matrix transformation and statistical convergence. Linear Algebra Appl. 2007, 420: 377-387. 10.1016/j.laa.2006.07.021MathSciNetView ArticleGoogle Scholar
  41. Mursaleen M, Çakan C, Mohiuddine SA, Savaş E: Generalized statistical convergence and statistical core of double sequences. Acta Math. Sin. Engl. Ser. 2010, 26: 2131-2144. 10.1007/s10114-010-9050-2MathSciNetView ArticleGoogle Scholar
  42. Kumar V, Mursaleen M:On ( λ , μ ) -statistical convergence of double sequences on intuitionistic fuzzy normed spaces. Filomat 2011, 25(2):109-120. 10.2298/FIL1102109KMathSciNetView ArticleGoogle Scholar
  43. Farah I Mem. Amer. Math. Soc. 148. Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers 2000.Google Scholar
  44. Caserta A, Kočinac LDR: On statistical exhaustiveness. Appl. Math. Lett. 2012, 25: 1447-1451. 10.1016/j.aml.2011.12.022MathSciNetView ArticleGoogle Scholar
  45. Caserta A, Di Maio G, Kočinac LDR: Statistical convergence in function spaces. Abstr. Appl. Anal. 2011., 2011: Article ID 420419Google Scholar
  46. Mohiuddine SA: An application of almost convergence in approximation theorems. Appl. Math. Lett. 2011, 24: 1856-1860. 10.1016/j.aml.2011.05.006MathSciNetView ArticleGoogle Scholar
  47. Mohiuddine SA, Alotaibi A: Statistical convergence and approximation theorems for functions of two variables. J. Comput. Anal. Appl. 2013, 15(2):218-223.MathSciNetGoogle Scholar
  48. Mohiuddine SA, Alotaibi A, Mursaleen M:Statistical summability ( C , 1 ) and a Korovkin type approximation theorem. J. Inequal. Appl. 2012., 2012: Article ID 172Google Scholar
  49. Srivastava HM, Mursaleen M, Khan A: Generalized equi-statistical convergence of positive linear operators and associated approximation theorems. Math. Comput. Model. 2012, 55: 2040-2051. 10.1016/j.mcm.2011.12.011MathSciNetView ArticleGoogle Scholar

Copyright

© Mohiuddine et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement