Open Access

A new variant of statistical convergence

  • Syed Abdul Mohiuddine1,
  • Abdullah Alotaibi1 and
  • Mohammad Mursaleen2Email author
Journal of Inequalities and Applications20132013:309

DOI: 10.1186/1029-242X-2013-309

Received: 27 March 2013

Accepted: 13 June 2013

Published: 3 July 2013

Abstract

In this paper we study the notion of statistical ( A , λ ) -summability, which is a generalization of statistical A-summability. We study here many other related concepts and its relations with statistical convergence and λ-statistical convergence and provide some interesting examples.

Keywords

density statistical convergence de la Vallée-Poussin regular matrix

1 Introduction and preliminaries

The concept of statistical convergence was first introduced by Fast [1]. In 1953 the concept arose as an example of convergence in density as introduced by Buck [2]. Schoenberg [3] studied statistical convergence as a summability method and Zygmund [4] established a relation between it and strong summability. This idea has grown a little faster after the papers of Šalát [5] , Fridy [6] , Connor [7, 8], Kolk [9], Mursaleen [10], Mursaleen and Edely [11, 12], Mursaleen and Mohiuddine [1317] and many others. Its various generalizations, extensions and variants have been studied by various authors so far. For example, lacunary statistical convergence [18], λ-statistical convergence [10, 1921], A-statistical convergence [9], statistical summability ( C , 1 ) [2224]; statistical λ-summability [25], statistical lacunary summability [26], statistical A-summability [27]etc. For more details, related concepts and applications, we refer to [2841] and references therein. Here we define the notion of statistical ( A , λ ) -summability as a λ-statistical convergence of A-transform of x and prove some results on some related sets of sequences. The results of this paper extend several ones obtained up to now and establish several inclusion relations, implications and other properties.

Let K N , the set of natural numbers. Then the natural density of K is defined by
δ ( K ) = lim n 1 n | { k n : k K } |

if the limit exists, where the vertical bars denote the cardinality of the enclosed set.

The idea of λ-statistical convergence was introduced in [10] as follows:

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 ] .

Let K N . Then
δ λ ( K ) = lim n 1 λ n | { n λ n + 1 j n : j K } |

is said to be λ-density of K.

In case λ n = n , λ-density reduces to the natural density. Also, since ( λ n / n ) 1 , δ ( K ) δ λ ( K ) for every K N .

A sequence x = ( x k ) is said to be λ-statistically convergent to L if for every ϵ > 0 the set K ϵ : = { k N : | x k L | ϵ } has λ-density zero, i.e., δ λ ( K ϵ ) = 0 . That is,
lim n 1 λ n | { n λ n + 1 j n : | x k L | ϵ } | = 0 .

In this case we write st λ - lim x = L .

Let A = ( a n k ) be an infinite matrix of real or complex numbers and x = ( x k ) be a sequence of real or complex numbers. Then we write A n ( x ) = k = 1 a n k x k , which is called the A-transform of the sequence x = ( x k ) whenever the series on the right converges for each n = 1 , 2 ,  .

We assume throughout this paper that the symbols ω and c denote the spaces of all sequences (real or complex numbers) and the space of all convergent sequences, respectively. Let X and Y be two nonempty subsets of the space ω. If x X implies A x = ( A n ( x ) ) Y , then we say that A defines a matrix transformation from X into Y, and we denote by ( X , Y ) the class of matrices A which transform X into Y. By ( X , Y ) reg we denote the subset of ( X , Y ) for which limit or sum is preserved.

A matrix A = ( a n k ) is said to be conservative if A x c for x = ( x k ) c , and we denote this by A ( c , c ) .

A matrix A = ( a n k ) is said to be regular if it is conservative and lim A x = lim x , and we denote this by A ( c , c ) reg .

The following are well-known Silverman-Toeplitz [42] conditions for the regularity of A.

A matrix A = ( a n k ) is regular, i.e., A ( c , c ) reg if and only if

(i) sup n k | a n k | < ;

(ii) lim n a n k = 0 , for each k;

(iii) lim n k a n k = 1 .

Let A = ( a i j ) be a non-negative regular matrix. A sequence x is said to be statistically A-summable to L if, for every ϵ > 0 , δ ( { i n : | y i L | ϵ } ) = 0 , i.e.,
lim n 1 n | { i n : | y i L | ϵ } | = 0 ,

where y i = A i ( x ) . Thus x is statistically A-summable to L if and only if Ax is statistically convergent to L. In this case we write L = ( A ) st - lim x = st - lim A x .

2 Statistical ( A , λ ) -summability

In [43], Malafosse and Rakočević presented the following definition of statistically ( A , λ ) -summable.

Definition 2.1 A sequence x is said to be statistically ( A , λ ) -summable to L if for every ϵ > 0 , δ λ ( { n λ n + 1 i n : | y i L | ϵ } ) = 0 , i.e.,
lim n 1 λ n | { n λ n + 1 i n : | y i L | ϵ } | = 0 .

Thus x is statistically ( A , λ ) -summable to L if and only if Ax is λ-statistically convergent to L. In this case we write L = ( A , λ ) st - lim x = st λ - lim A x . By ( A , λ ) st we denote the set of all statistically ( A , λ ) -summable sequences.

We define the following.

Definition 2.2 A sequence x = ( x k ) is said to be strongly ( A , λ q ) -convergent ( 0 < q < ) to the limit L if lim n 1 λ n i I n | y i L | q = 0 , and we write it as x k L [ A , λ ] q . In this case L is called the [ A , λ ] q -limit of x.

Remarks 2.3

(i) If A = I (the unit matrix), then the statistical ( A , λ ) -summability is reduced to the λ-statistical convergence.

(ii) If λ n = n , then the statistical ( A , λ ) -summability is reduced to the statistical A-summability.

(iii) If λ n = n and
a i k = { 1 i + 1 , 0 k i , 0 , otherwise ,

then the statistical ( A , λ ) -summability is reduced to the statistical ( C , 1 ) -summability due to Moricz [22].

(iv) If λ n = n and
a i k = { p k P i , 0 k i , 0 , otherwise ,
then the statistical ( A , λ ) -summability is reduced to the statistical ( N ¯ , p ) -summability due to Moricz and Orhan [44], where p = ( p k ) is a sequence of nonnegative numbers such that p 0 > 0 and
P i = k = 0 i p k ( i ) .
(v) If λ n = n and
a i k = { 1 k l i , 0 k i , 0 , otherwise ,

where l i = k = 0 i 1 ( k + 1 ) , then the statistical ( A , λ ) -summability is reduced to the statistical ( H , 1 ) -summability due to Moricz [45].

3 Main results

In this section, we establish the relation between statistical ( A , λ ) -summability and A-statistical convergence.

Theorem 3.1 If a bounded sequence is A-statistically convergent to and lim inf n λ n n > 0 , then it is A summable to , statistically A-summable to , and hence statistically ( A , λ ) -summable to but not conversely.

Proof Let x be bounded and A-statistically convergent to L, and K ϵ = { k n : | x k L | ϵ } . Then
| A n ( x ) L | = | k = 1 a n k ( x k L ) + L ( k = 1 a n k 1 ) | k = 1 a n k | x k L | + | L | | k = 1 a n k 1 | = k K ϵ a n k | x k L | + k K ϵ a n k | x k L | + | L | | k = 1 a n k 1 | sup k | x k L | k K ϵ a n k + ϵ k K ϵ a n k + | L | | k = 1 a n k 1 | .
By using the definition of A-statistical convergence and the conditions of regularity of A, we get
lim | A n ( x ) L | = 0 since  ϵ  was arbitrary ,

and hence st - lim | A n ( x ) L | = 0 , i.e., x is statistically A-summable to L. Now, using Theorem 3.1 of [10], we get st λ - lim | A n ( x ) L | = 0 , i.e., x is statistically ( A , λ ) -summable to L.

To see that the converse does not hold, we construct the following example.

Let λ n = n and A be a Cesàro matrix, i.e.,
a n k = { 1 n + 1 , 0 n k , 0 , otherwise .
Let
x k = { 1 , if  k  is odd , 0 , if  k  is even .

Then x is A-summable to 1 / 2 (and hence statistically ( A , λ ) -summable to 1 / 2 ) but not A-statistically convergent.

This completes the proof of the theorem. □

Theorem 3.2 If lim sup n ( n λ n ) < and x is statistically ( A , λ ) -summable to L, then x is statistically A-summable to L.

Proof Let lim sup n ( n λ n ) < . Then there exists M > 0 such that n λ n M for all n. Since 1 n 1 λ n and
{ 1 i n : | y i L | ε } { i I n : | y i L | ε } { 1 i n λ n : | y i L | ε } ,
we have
1 n | { 1 i n : | y i L | ε } | 1 λ n | { 1 i n : | y i L | ε } | 1 λ n | { i I n : | y i L | ε } | + 1 λ n | { i n λ n : | y i L | ε } | 1 λ n | { i I n : | y i L | ε } | + M λ n .

Now, taking the limit as n , we get the desired result. □

Theorem 3.3 Statistical ( A , λ ) -summability implies statistical A-summability if and only if
lim inf n λ n n > 0 .
(3.1)
Proof For ε > 0 , we have
{ i I n : | y i L | ε } { i n : | y i L | ε } .
Therefore
1 n | { i n : | y i L | ε } | 1 n | { i I n : | y i L | ε } | λ n n 1 λ n | { i I n : | y i L | ε } | .

Taking the limit as n and using (3.1), we get that statistical ( A , λ ) -summability implies statistical A-summability.

Conversely, suppose that
lim inf n λ n n = 0 .
Choose a subsequence ( n ( j ) ) j 1 such that λ n ( j ) n ( j ) < 1 j . Define a sequence x = ( x k ) k 1 such that
y i = { 1 , for  i I n ( j ) , j = 1 , 2 , 3 , , 0 , otherwise .

Then, as in Theorem 3.1 of [10], we get that y = ( y i ) is not λ-statistically convergent, i.e., x is not statistically ( A , λ ) -summable. Hence (3.1) is necessary.

This completes the proof of the theorem. □

Theorem 3.4 (a) If 0 < q < and a sequence x = ( x k ) is strongly ( A , λ q ) -convergent to the limit L, then x is statistically ( A , λ ) -convergent to L.

(b) If x = ( x k ) is bounded and statistically ( A , λ ) -convergent to L, then x k L [ A , λ ] q .

Proof (a) It follows easily from the following:
1 λ n i I n | y i L | q ε q λ n | { i I n : | y i L | ε } | .
The following example shows that the inclusion is proper. Let x = ( x n ) n 1 be such that its A-transform is given by
y i = { i , for  n [ λ n ] + 1 i n , 0 , otherwise .
Then A x and for 0 < ε 1 ,
1 λ n | { i I n : | y i 0 | ε } | = [ λ n ] λ n 0 ( n ) ,
i.e., x is statistically ( A , λ ) -convergent to 0. But
1 λ n i I n | y i 0 | q 0 ,

i.e., x is not strongly ( A , λ q ) -convergent to the limit 0.

(b) Suppose x = ( x k ) is bounded and statistically ( A , λ ) -convergent to L. Then | x k L | M for all k, where M > 0 . For ε > 0 , we have
1 λ n k I n | y i L | q = 1 λ n i I n | y i L | q ϵ | y i L | q + 1 λ n i I n | y i L | q < ϵ | y i L | q M λ n | { i I n : | y i L | ε } | + ε q .

Hence x k L [ A , λ ] q if x is statistically ( A , λ ) -convergent to L.

This completes the proof of the theorem. □

Declarations

Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (334/130/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University
(2)
Department of Mathematics, Aligarh Muslim University

References

  1. Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244.MathSciNetMATHGoogle Scholar
  2. Buck RC: Generalized asymptotic density. Am. J. Math. 1953, 75: 335–346. 10.2307/2372456MathSciNetView ArticleMATHGoogle Scholar
  3. Schoenberg IJ: The integrability of certain functions and related summability methods. Am. Math. Mon. 1959, 66: 361–375. 10.2307/2308747MathSciNetView ArticleMATHGoogle Scholar
  4. Zygmund A: Trigonometric Series. Cambridge University Press, Cambridge; 1959.MATHGoogle Scholar
  5. Šalát T: On statistically convergent sequences of real numbers. Math. Slovaca 1980, 30: 139–150.MathSciNetMATHGoogle Scholar
  6. Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.MathSciNetView ArticleMATHGoogle Scholar
  7. Connor J: The statistical and strong p -Cesàro convergence of sequences. Analysis 1988, 8: 47–63.MathSciNetView ArticleMATHGoogle Scholar
  8. Connor J: On strong matrix summability with respect to a modulus and statistical convergence. Can. Math. Bull. 1989, 32: 194–198. 10.4153/CMB-1989-029-3MathSciNetView ArticleMATHGoogle Scholar
  9. Kolk E: Matrix summability of statistically convergent sequences. Analysis 1993, 13: 77–83.MathSciNetView ArticleMATHGoogle Scholar
  10. Mursaleen M: λ -statistical convergence. Math. Slovaca 2000, 50: 111–115.MathSciNetMATHGoogle Scholar
  11. Mursaleen M, Edely OHH: Statistical convergence of double sequences. J. Math. Anal. Appl. 2003, 288: 223–231. 10.1016/j.jmaa.2003.08.004MathSciNetView ArticleMATHGoogle Scholar
  12. Mursaleen M, Edely OHH: Generalized statistical convergence. Inf. Sci. 2004, 162: 287–294. 10.1016/j.ins.2003.09.011MathSciNetView ArticleMATHGoogle Scholar
  13. 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 ArticleMATHGoogle Scholar
  14. Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233(2):142–149. 10.1016/j.cam.2009.07.005MathSciNetView ArticleMATHGoogle Scholar
  15. Mursaleen M, Mohiuddine SA: On ideal convergence of double sequences in probabilistic normed spaces. Mathem. Rep. (Buchar.) 2010, 12(4):359–371.MathSciNetMATHGoogle Scholar
  16. 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 ArticleMATHGoogle Scholar
  17. Mursaleen M, Mohiuddine SA: On ideal convergence in probabilistic normed spaces. Math. Slovaca 2012, 62(1):49–62. 10.2478/s12175-011-0071-9MathSciNetView ArticleMATHGoogle Scholar
  18. Fridy JA, Orhan C: Lacunary statistical convergence. Pac. J. Math. 1993, 160: 43–51. 10.2140/pjm.1993.160.43MathSciNetView ArticleMATHGoogle Scholar
  19. 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 ArticleMATHGoogle Scholar
  20. Savaş E, Mohiuddine SA: λ ¯ -statistically convergent double sequences in probabilistic normed spaces. Math. Slovaca 2012, 62(1):99–108. 10.2478/s12175-011-0075-5MathSciNetMATHGoogle Scholar
  21. Mohiuddine SA, Alotaibi A, Mursaleen M: Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces. Adv. Differ. Equ. 2013., 2013: Article ID 66Google Scholar
  22. Moricz F:Tauberian conditions under which statistical convergence follows from statistical summability ( C , 1 ) . J. Math. Anal. Appl. 2002, 275: 277–287. 10.1016/S0022-247X(02)00338-4MathSciNetView ArticleMATHGoogle Scholar
  23. 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
  24. Mohiuddine SA, Alotaibi A:Korovkin second theorem via statistical summability ( C , 1 ) . J. Inequal. Appl. 2013., 2013: Article ID 149Google Scholar
  25. Mursaleen M, Alotaibi A: Statistical summability and approximation by de la Vallée-Poussin mean. Appl. Math. Lett. 2011, 24: 320–324. (Erratum: Appl. Math. Lett. 25, 665 (2012)) 10.1016/j.aml.2010.10.014MathSciNetView ArticleMATHGoogle Scholar
  26. Mursaleen M, Alotaibi A: Statistical lacunary summability and a Korovkin type approximation theorem. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 2011, 57(2):373–381. 10.1007/s11565-011-0122-8MathSciNetView ArticleMATHGoogle Scholar
  27. Edely OHH, Mursaleen M: On statistical A -summability. Math. Comput. Model. 2009, 49: 672–680. 10.1016/j.mcm.2008.05.053MathSciNetView ArticleMATHGoogle Scholar
  28. Alotaibi A, Mursaleen M: A -statistical summability of Fourier series and Walsh-Fourier series. Appl. Math. Inf. Sci. 2012, 6(3):535–538.MathSciNetGoogle Scholar
  29. Freedman AR, Sember JJ: Densities and summability. Pac. J. Math. 1981, 95: 293–305. 10.2140/pjm.1981.95.293MathSciNetView ArticleMATHGoogle Scholar
  30. Fridy JA, Miller HI: A matrix characterization of statistical convergence. Analysis 1991, 11: 59–66.MathSciNetView ArticleMATHGoogle Scholar
  31. Mohiuddine SA, Alghamdi MA: Statistical summability through a lacunary sequence in locally solid Riesz spaces. J. Inequal. Appl. 2012., 2012: Article ID 225Google Scholar
  32. Mohiuddine SA, Aiyub M: Lacunary statistical convergence in random 2-normed spaces. Appl. Math. Inf. Sci. 2012, 6(3):581–585.MathSciNetGoogle Scholar
  33. 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
  34. 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
  35. 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 ArticleMATHGoogle Scholar
  36. Mohiuddine SA, Savas E: Lacunary statistical convergent double sequences in probabilistic normed spaces. Ann. Univ. Ferrara 2012, 58: 331–339. 10.1007/s11565-012-0157-5MathSciNetView ArticleMATHGoogle Scholar
  37. Mohiuddine SA, Şevli H, Cancan M: Statistical convergence in fuzzy 2-normed space. J. Comput. Anal. Appl. 2010, 12(4):787–798.MathSciNetMATHGoogle Scholar
  38. Mohiuddine SA, Şevli H, Cancan M: Statistical convergence of double sequences in fuzzy normed spaces. Filomat 2012, 26(4):673–681. 10.2298/FIL1204673MMathSciNetView ArticleMATHGoogle Scholar
  39. Belen C, Mohiuddine SA: Generalized weighted statistical convergence and application. Appl. Math. Comput. 2013, 219: 9821–9826. 10.1016/j.amc.2013.03.115MathSciNetView ArticleMATHGoogle Scholar
  40. Mohiuddine SA, Alotaibi A: Statistical convergence and approximation theorems for functions of two variables. J. Comput. Anal. Appl. 2013, 15(2):218–223.MathSciNetMATHGoogle Scholar
  41. Mohiuddine SA, Hazarika B, Alotaibi A: Double lacunary density and some inclusion results in locally solid Riesz spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 507962Google Scholar
  42. Cooke RG: Infinite Matrices and Sequence Spaces. Macmillan & Co., London; 1950.MATHGoogle Scholar
  43. de Malafosse B, Rakočević V: Matrix transformation and statistical convergence. Linear Algebra Appl. 2007, 420: 377–387. 10.1016/j.laa.2006.07.021MathSciNetView ArticleMATHGoogle Scholar
  44. Moricz F, Orhan C: Tauberian conditions under which statistical convergence follows from statistical summability by weighted means. Studia Sci. Math. Hung. 2004, 41(4):391–403.MathSciNetMATHGoogle Scholar
  45. Moricz F: Theorems related to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences. Analysis 2004, 24: 127–145.MathSciNetView ArticleMATHGoogle Scholar

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