aa r X i v : . [ m a t h . F A ] M a r Note on f λ -statistical convergence Stuti Borgohain ∗ and Ekrem Savas¸Istanbul Commerce University, 34840 Istanbul, TurkeyE-mail : [email protected], [email protected]. Abstract:
In this article, we study about the λ -statistical convergence with respect to the densityof moduli and find some results related to statistical convergence as well. Also we introduce theconcept of f λ -summable sequence and try to investigate some relation between the f λ -summabilityand module λ -statistical convergence. The idea of statistical convergence of real sequences is the extension of conver-gence of real sequences. Initially the statistical convergence for single sequenceswas introduced by Fast Fast [18] in 1951 and Schoenberg [23] in 1959, studiedstatistical convergence as a summability method and listed some of elementaryproperties of statistical convergence as well. Both of these authors noted that ifbounded sequence is statistically convergent, then it is Cesaro summable. Moreinvestigations have been studied in the direction of topological spaces, statisticalCauchy condition in uniform spaces, Fourier analysis, ergodic theory, numbertheory, measure theory, trigonometric series, turnpike theory and Banach spaces.Statistical convergence turned out to be one of the most active areas of research insummability theory after the works of Fridy [25] and Salat [28]. Recently, Savas[5] introduced the generalized double statistical convergence in locally solid Rieszspaces. Moreover, Mursaleen [26] introduced the concept of λ -statistically con-vergence by using the idea of ( V , λ )-summability to generalize the concept ofstatistical convergence. For more interesting investigations concerning statisticalconvergence one may consult the papers of Cakalli [16], Miller and Orhan [17] The work of the first author was carried during her one month visit to Istanbul TicaretUniversity, Turkey, in November’2015 ? ]) and others.The concept of density for sets of natural numbers with respect to the modulusfunction was introduced by A. Aizpuru et al [1] in 2014. They studied and charac-terized the generalization of this notion of f -density with statistical convergenceand proved that ordinary convergence is equivalent to the module statistical con-vergence for every unbounded modulus function. Also, in [2], A. Aizpuru andhis team, they worked on double sequence spaces for the results of f -statisticalconvergence by using unbounded modulus function. The concept was furthergeneralized and characterized by Savas and Borgohain [[ ? ], [6]] .The notion of statistical convergence depends on the idea of asymtotic densityof subsets of the set N of natural numbers. A subset A of N is said to have naturaldensity δ ( A ) if δ ( A ) = lim n →∞ n n X k = χ A ( k ) . where χ A is the characteristic function of A .We mean a sequence ( x t ) to be statistically convergent to L , if for any ξ > δ { ( t ∈ N : | x t − L | ≥ ξ ) } = . Analogously, ( x t ) is said to be statistically Cauchy if foreach ξ > n ∈ N there exists an integer q ≥ n such that δ ( { t ∈ N : k x t − x q k <ξ } ) = . Nakano [19] introduced the notion of a modulus function whereas Ruckle [29]and Maddox [21] have introduced and discussed some properties of sequencespaces defined by using a modulus function. A modulus function is a function f : R + → R + which satisfies:1. f ( x ) = x = f ( x + y ) ≤ f ( x ) + f ( y ) for every x , y ∈ R + .3. f is increasing.4. f is continuous from the right at 0.2ater on modulus function have been discussed in ([20, 10, 11, 12, 14]) and others.In this paper, we study the density on moduli with respect to the λ -statisticalconvergence. We also investigate some results on the new concept of f λ -statisticalconvergence with the ordinary convergence. Also we find out some new con-cepts on f λ -summability theory and try to find out new results related to the f λ -summable sequences and f λ -statistical convergent sequence. By density of moduli of a set A ⊆ N , we mean δ f ( A ) = lim u f ( | A ( u ) | ) f ( u ) , (in case thislimit exists )where f is an unbounded modulus function.Let ( x t ) be a sequence in X ( X is a normed space). If for each ξ > A = { t ∈ N : k x t − L k > ξ } has f -density zero, then it is said that the f -statisticallimit of ( x t ) is L ∈ X , and we write it as f -stlim x t = L .Observe that δ ( A ) = − δ ( N \ A ).Let us assume that A ⊆ N and A has f -density zero. For every u ∈ N we have f ( u ) ≤ f ( | A ( u ) | ) + f ( | ( N \ A )( u ) | ) and so1 ≤ f ( | A ( u ) | ) f ( u ) + f ( | ( N \ A )( u ) | ) f ( u ) ≤ f ( | A ( u ) | ) f ( u ) + f -density of ( N \ A ) is one.Let λ = ( λ n ) be a non-decreasing sequence of positive numbers such that , λ = , λ n + ≤ λ n + λ n → ∞ as n → ∞ . Note: The collection of all such sequences λ will be denoted by Λ .A sequence ( x t ) of real numbers is said to be λ -statistically convergent to L iffor any ξ >
0, lim n →∞ λ n |{ t ∈ I n : | x t − L | ≥ ξ }| = , I n = [ n − λ n + , n ] and | A | denotes the cardinality of A ⊂ N (refer [26]).More investigations in this direction and more applications of ideals can befound in [5, 8, 9, 15] where many important references can be found.We define f λ -statistical convergence as:lim n →∞ f ( | A ( n ) | ) f ( λ n ) = δ f λ ( A ) = lim n →∞ f ( | A ( n ) | ) f ( λ n ) . The set of all f λ -statistically convergent sequences is denoted by S f λ .A sequence x = ( x t ) is said to be strongly f λ -summable to the limit L iflim n f ( λ n ) X t ∈ I n f ( | x t − L | ) = x t → L [ f λ ]. In this case, L is called the f λ -limit of x . We denotethe class of all f λ -summable sequences as w f λ . Maddox [22] showed the existence of an unbounded modulus f for which thereexists a positive constant c such that f ( xy ) ≥ c f ( x ) f ( y ) for all x ≥ , y ≥ Theorem 3.1.
Let f be an unbounded modulus such that there is a positive constantc such that f ( xy ) ≥ c f ( x ) f ( y ) for all x ≥ , y ≥ and lim u →∞ f ( u ) u > . A strongly λ -summable sequence ( x t ) is also a f λ -statistically convergent sequence. Moreover, for abounded sequence ( x t ) , f λ -statistically convergence implies strongly λ -summability.Proof. By the definition of modulus function, we have for any sequence x = ( x t )and ξ > n X t = f ( | x t − L | ) ≥ f n X t = f ( | x t − L | ≥ f ( |{ t ≤ n : | x t − L | ≥ ξ }| ξ ) ≥ c f ( |{ t ≤ n : | x t − L | ≥ ξ }| ) f ( ξ )4ince x = ( x t ) is strongly λ -summable sequence, so1 λ n n X t = f ( | x t − L | ) ≥ c f ( |{ t ≤ n : | x t − L | ≥ ξ }| ) f ( ξ ) λ n = c f ( |{ t ≤ n : | x t − L | ≥ ξ }| ) f ( ξ ) f ( λ n ) λ n f ( λ n )By using the fact that lim u →∞ f ( u ) u > x is λ -statistical w.r.t. f , it can besummarized that x is f λ -statistcal convergent and this completes the proof of thetheorem. (cid:3) Theorem 3.2.
For a strongly λ -summable or statistically f λ -convergent sequence x = ( x t ) to L, there is a convergent sequence y and a f λ -statistically null sequence z such that y isconvergent to L, x = y + z and lim n f ( |{ t ∈ I n : z t , }| ) f ( λ n ) = . Moreover, if x is bounded,then y and z both are bounded. Note: Here f be an unbounded modulus such that there is a positive constant c such that f ( xy ) ≥ c f ( x ) f ( y ) for all x ≥ , y ≥ u →∞ f ( u ) u > Proof.
From the previous theorem, we have x is strongly λ -summable to L implies x is f λ -statistically convergent to L . Choose a strictly increasing sequence ofpositive integers M < M < M ... such that,1 f ( λ n ) f (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:26) t ∈ I n : | x t − L | ≥ d (cid:27)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) < d , for n > N d , where N = N < t < N , let us set z t = y t = x t .Let d ≥ N d < t ≤ N d + , we set, y t = x t , z t = | x t − L | < d ; L , z t = x t − L if | x t − L | ≥ d . Clearly, x = y + z and y and z are bounded, if x is bounded.Observe that | y t − L | < ξ for t > N d and | y t − L | = | x t − L | < ξ if | x t − L | < d .5hich follows that | y t − L | = | L − L | = | x t − L | > d .Hence, for ξ arbitrary, we get lim t y t = L .Next, we observe that, f ( |{ t ∈ I n : z t , }| ) ≥ f ( |{ t ∈ I n : | z t | ≥ ξ }| ), for any naturalnumber n and ξ >
0. Hence,lim n f ( |{ t ∈ I n : z t , }| ) f ( λ n ) = z is f λ -statistically null.We now show that if β > d ∈ N such that d < β , then f ( |{ t ∈ I n : z t , }| ) < β for all n > N d .If N d < t ≤ N d + , then z t , | x t − L | > d . It follows that if N p < t ≤ N p + ,then, { t ∈ I n : z t , } ⊆ ( t ∈ I n : | x t − L | > p ) . Consequently,1 f ( λ n ) f ( |{ t ∈ I n : z t , }| ) ≤ f ( λ n ) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( t ∈ I n : | x t − L | > p )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! < p < d < β, if N p < n ≤ N p + and p > d .That is, lim n f ( λ n ) f ( |{ t ∈ I n : z t , }| ) = (cid:3) Corollary 3.1.
Let f , be unbounded moduli, X a normed space, ( x t ) a sequence in Xand x , y ∈ X. We have1. A f λ -statistical convergent sequence is also statistically convergent to the samelimit. . The f λ -statistical limit is unique whenever it exists.3. Let f λ -stat lim x t = x and f λ -stat lim y t = y, then, • f λ -stat lim( x t ± y t ) = x ± y • f λ -stat lim α x t = α x , α ∈ R .
4. If f -st lim x t = x and -st lim x t = y then x = y, i.e two di ff erent methods ofstatistical convergence are always compatible. Theorem 3.3.
Let ( x t ) be a sequence in a normed space X. For an unbounded modulusf , ( x t ) is f λ -statistically convergent to x if and only if there exists T ⊆ N such thatf λ -density of T is zero and ( x t ) has λ -summable to x.Proof. Let us assume that V z = t ∈ N : X t k x t − x k > z , for every z ∈ N suchthat V z ⊂ V z + and lim n f ( | V z ( n ) | ) f ( λ n ) = f ( | V z ( n ) | ) f ( λ n ) ≤ z whenever n ≥ i z .Note: Here i < i < i ... is a strictly increasing sequence of positive numberssuch that i z ∈ V z .Again set T = ∪ z ∈ N ([ i z , i z + ) ∩ V z ). Then for every n ≥ i there exists z ∈ N suchthat i z ≤ n ≤ i z + and if m ∈ T ( n ) then m < i z + , which implies m ∈ V z . Therefore T ( n ) ⊆ V z ( n ) and thus, f ( | T ( n ) | ) f ( λ n ) ≤ f ( | V z ( n ) | ) f ( λ n ) ≤ z which follows that T has f λ -density zero.For ξ > z ∈ N such that z < ξ , we have n ∈ N \ T and n ≥ i z for whichthere exists q ≥ z with i q ≤ n ≤ i q + and this implies n < V q , so,1 λ n X t k x t − x k ≤ q ≤ z < ξ. x t ) has λ -summable to x .Conversely, let us assume that for T ∈ N , lim t ∈ N \ T λ n k x t − x k = n ∈ I n and T has f λ -density zero. For ξ >
0, there exists t ∈ N such that if t > t and t ∈ N \ T then 1 λ n X t k x t − x k ≤ ξ. This implies, t ∈ N : X t k x t − x k > ξ ⊆ T ∪ { , .. t } and then δ f λ t ∈ N : X t k x t − x k > ξ = (cid:3) Definition 3.1.
The sequence ( x t ) is f λ -statistically Cauchy if for every ξ > there existsQ ∈ N such that δ f λ t ∈ N : X t k x t − x Q k > ξ = . Corollary 3.2.
A sequence is f λ -statistically convergent implies that it is f λ -statisticallyCauchy. The converge is true if the space is complete and this result is a particular case offilter convergence. Theorem 3.4.
For the Banach space X where f be an unbounded modulus, f λ -statisticallyCauchy sequence is f λ -statistically convergent sequence.Proof. For every k ∈ N , let x Q k be such that δ f λ t ∈ N : X t k x t − x Q k k > k = . Consider the set J k = ∩ i ≤ k B ( x Q i , i ), then for each k ∈ N we have diam( J k ) ≤ k R k = { t ∈ N : x t < J k } so that R k = ∪ i ≤ k t ∈ N : X t k x t − x Q i k > i whichimplies f λ -density of R k is zero.Following the proof of the previous theorem, we get for t ≥ r k then f ( | R k ( t ) | ) f ( λ t ) ≤ k , r k ∈ R k .Considering D = ∪ k ∈ N ([ r k , r k + ) ∩ R k ), we get f λ -density of D is zero.Since ∩ k ∈ N J k has exactly one element, say x , due to the completeness of X . Wehave to prove that lim t ∈ N \ D λ n k x t − x k = ξ >
0, let us choose i ∈ N such that i < ξ . If t ≥ r j and t ∈ N \ D then thereexists k ≥ i such that r k ≤ t < r k + and then t < R k , which implies x t ∈ J k and thus,1 λ n X t k x t − x k ≤ k ≤ i < ξ. This completes the proof of the theorem. (cid:3)
Corollary 3.3.
A sequence which is f λ -statistically convergent for each module f , is alsoconvergent in ordinary sense, i.e. if B ⊆ N is infinite, then there exists an unboundedmodule f such that f λ -density of B is one. Theorem 3.5.
Let ( x t ) be a sequence in X. If for every unbounded modulus f there existsf λ -st lim x t then all these limits are the same x ∈ X and ( x t ) is also λ -summable to x in thenorm topology.Proof. It is proved that for f , two unbounded moduli, the f λ -statistical limit isunique whenever it exists.Let X a normed space, ( x t ) a sequence in X and P , Q ∈ X and if the uniquenessis false that lim x t = P , there exists ξ > = t ∈ N : X p k x t − P k > ξ is infinite.Now, by choosing an unbounded modulus f which will satisfy δ f λ ( B ) =
1, thenthis clearly contradicts the assumption that f λ -stlim x t = x , which completes theproof of the Theorem. (cid:3) Theorem 3.6. If λ ∈ Λ with lim n →∞ in f nf ( λ n ) > , then f λ -statistically convergentsequences are f -statistical convergent.Proof. For given ξ >
0, we have, { t ≤ n : | x t − L | ≥ ξ } ⊃ { t ∈ I n : | x t − L | ≥ ξ } . Therefore,1 f ( λ n ) f ( |{ t ≤ n : | x t − L | ≥ ξ }| ) ≥ f ( λ n ) f ( |{ t ∈ I n : | x t − L | ≥ ξ }| ) ≥ f ( n ) n nf ( λ n ) 1 f ( n ) f ( |{ t ∈ I n : | x t − L | ≥ ξ }| )Taking the limit as n → ∞ and using the fact that lim n f ( n ) n >
0, we get ( x t ) is f λ -statistical convergent implies ( x t ) is f -statistical convergent. (cid:3) Theorem 3.7.
Let λ ∈ Λ be such that lim n nf ( λ n ) = . Then, f -statistical convergence isf λ -statistical convergence.Proof. Let β > n nf ( λ n ) =
1, we can choose m ∈ N such that, f ( n − λ n + f ( n ) < β for all n ≥ m .Now observe that, 10 f ( n ) f ( |{ t ≤ n : | x t − L | ≥ ξ }| ) = f ( n ) f ( |{ t < n − λ n + | x t − L | ≥ ξ }| ) + f ( n ) f ( |{ t ∈ I n : | x t − L | ≥ ξ }| ) < f ( n − λ n + f ( n ) + f ( n ) f ( |{ t ∈ I n : | x t − L | ≥ ξ }| ) < β + f ( λ n ) f ( |{ t ∈ I n : | x t − L | ≥ ξ }| )Taking limit as n → ∞ , we get, ( x t ) is f -statistically convergent is f λ -statisticallyconvergent. (cid:3) Theorem 3.8.
Let λ = ( λ n ) and µ = ( µ n ) be two sequences in Λ such that λ n ≤ µ n forall n ∈ N n .1. If lim inf n →∞ λ n µ n > then S f µ ⊆ S f λ .2. If lim n →∞ µ n λ n = then S f λ ⊆ S f µ .Proof.
1. Suppose that λ n ≤ µ n for all n ∈ N n which implies that f ( λ n ) ≤ f ( µ n ).Given lim inf n →∞ λ n µ n >
0, then with the property of modulus function, we getlim inf n →∞ f ( λ n ) f ( µ n ) >
0. Also, since λ n ≤ µ n for all n ∈ N n , so I n ⊂ J n where I n = [ n − λ n + , n ] and J n = [ n − µ n + , n ].Now, for ξ >
0, we can write, { t ∈ J n : | x t − L | ≥ ξ } ⊃ { t ∈ I n : | x t − L | ≥ ξ } and so, µ n |{ t ∈ J n : | x t − L | ≥ ξ }| ≥ λ n µ n λ n |{ t ∈ I n : | x t − L | ≥ ξ }| , for all n ∈ N n .Now, by using the definition of modulus function and satisfying lim inf n →∞ λ n µ n > n → ∞ , we get S f µ ⊆ S f λ . (cid:3) roof.
2. Let S f λ − lim x t = x and lim n →∞ µ n λ n = I n ⊂ J n , for ξ >
0, we write,1 µ n |{ t ∈ J n : | x t − L | ≥ ξ }| = µ n |{ n − µ n + ≤ k ≤ n − λ n : | x t − L | ≥ ξ }| + µ n |{ t ∈ I n : | x t − L | ≥ ξ }|≤ µ n − λ n µ n + µ n |{ t ∈ I n : | x t − L | ≥ ξ }|≤ µ n − λ n λ n + µ n |{ t ∈ I n : | x t − L | ≥ ξ }|≤ (cid:18) µ n λ n − (cid:19) + λ n |{ t ∈ I n : | x t − L | ≥ ξ }| , for all n ∈ N n . Since lim n →∞ µ n λ n = x = ( x t ) is S f µ -statistically convergent is S f λ -statistcially convergent sequence,so S f λ ⊆ S f µ . (cid:3) Corollary 3.4.
Let λ = ( λ n ) and µ = ( µ n ) be two sequences in Λ such that λ n ≤ µ n forall n ∈ N n . If lim inf n →∞ λ n µ n > , then w f µ ⊂ w f λ . Theorem 3.9.
Let λ n ≤ µ n for all n ∈ N n , then, if lim inf n →∞ λ n µ n > , then a stronglyw f µ -summable sequence is S f λ -statistically convergent sequence.Proof. For any ξ >
0, we have, X t ∈ J n f ( | x t − L | ) = X t ∈ J n , f ( | x t − L | ) ≥ ξ f ( | x t − L | ) + X t ∈ J n , f ( | x t − L | ) <ξ f ( | x t − L | ) ≥ X t ∈ I n , f ( | x t − L | ) ≥ ξ f ( | x t − L | ) + X t ∈ I n , f ( | x t − L | ) <ξ f ( | x t − L | ) ≥ X t ∈ I n , f ( | x t − L | ) ≥ ξ f ( | x t − L | ) ≥ f X t ∈ J n , f ( | x t − L | ) ≥ ξ | x t − L | ≥ f ( |{ t ∈ I n : | x t − L | ≥ ξ }| .ξ ) ≥ c f ( |{ t ∈ I n : | x t − L | ≥ ξ }| ) . f ( ξ )12nd so 1 µ n X t ∈ J n f ( | x t − L | ) ≥ µ n f ( |{ t ∈ I n : | x t − L | ≥ ξ }| ) f ( ξ ) ≥ λ n µ n λ n f ( |{ t ∈ I n : | x t − L | ≥ ξ }| ) f ( ξ )Since lim inf n →∞ λ n µ n > x t ) is strongly f λ -summable to L implies ( x t ) is S f λ statistically convergent to L .This completes the proof of the theorem. (cid:3) References [1] A. Aizpuru, M.C. List ´ a n-Garc´ i and F. Rambla-Barreno, Density by Mod-uli and Statistical Convergence, Quaestiones Mathematicae 2014: 1-6.http: // dx.doi.org / / ff erence Sequencespaces of Fuzzy Real numbers defined by Orlicz Function, Thai Journal ofMathematics , 11(2) (2013), 357-370.[3] E. Kolk, The statistical convergence in Banach spaces, A cta Et Commenta-tiones Univ. Tartuensis, 928(1991), 41-52.[4] E. Kolk, Matrix summability of statistically convergent sequences, Analysis,13(1993), 77-83.[5] E. Savas¸, On generalize ddouble statistical convergence in locally solid Rieszspaces, Miskolc Mathematical Notes, preprint.[6] E. Savas¸ and S. Borgohain , On strongly almost lacunary statistical A -convergence defined by a Musielak-Orlicz function. (accepted in Filomat)137] E. Savas¸ and S. Borgohain, Some new spaces of lacunary f -statistical A -convergent sequences of order α , Advancements in Mathematical Sciences,AIP Conf. Proc. 1676, 2015, 020086-1ˆa e “020086-8; doi: 10.1063 / I λ -statistically convergent sequences in topological groups, ActaEt Commentationes Universitatis Tartuensis De Mathematica, 18(1)(2014),33-38.[10] E. Savas¸, On generalized sequence spaces via modulus function, J. Inequal.Appl.,2014, 2014:101, 8 pp. 40A30.[11] E. Savas¸, On some new sequence spaces defined by infinite matrix and mod-ulus, Adv. Di ff erence Equ., 2013, 2013:274, 9 pp.[12] E. Savas¸ and R. F. Patterson, Double sequence spaces defined by a modulus.Math. Slovaca, 61(2), 245-256(2011).[13] E. Savas¸ and R. F. Patterson, Lacunary statistical convergence of multiplesequences, Appl. Math. Lett., 19(6), 527-534(2006).[14] E. Savas¸, On some generalized sequence spaces defined by a modulus, IndianJ. Pure Appl. Math. 30(5), 459-464(1999).[15] E. Savas¸, Strong almost convergence and almost λ -statistical convergence,Hokkaido Math. J., 29(3), 531-536(2000).[16] H. Cakalli and E. Savas¸, Statistical convergence of double sequences in topo-logical groups, J. Comput. Anal. Appl., 12(2), 421-426(2010).[17] H. I. Miller and C. Orhan, On almost convergent and statistically convergentsubsequences, Acta Math. Hungar., 93(1-2)(2001), 135-151.[18] H. Fast, Sur la convergence statistique, Colloq. Math. 2(1951), 241-244.[19] H. Nakano, Concave modulars, J. Math. Soc. Japan., 5 (1953), 29ˆa e “49.1420] I. Bala, V. K. Bhardwaj and E. Savas¸, Some sequence spaces defined by | A | summability and a modulus function in seminormed space, Thai J. Math.,11(3), 623-631(2013).[21] I. J. Maddox , Sequence Spaces Defined by a modulus, Mathematical Pro-ceedings of the Cambridge Philosophical Society, 100 (1986), 161-166.[22] I.J. Maddox, Inclusion between FK spaces and Kuttnerˆa’s theorem, Math.Proc. Camb. Philos. Soc., 101(1987), 523-527.[23] I.J. Schoenberg, The integrability of certain functions and related summabilitymethods, Amer. Math. Monthly, 66(1959), 361-375.[24] J. Connor and E. Savas¸, Lacunary statistical and sliding window convergencefor measurable functions, Acta Mathematica Hungarica, 145(2)(2015), 416-432.[25] J.A. Fridy, On statistical convergence, Analysis, 5(1985), 301-313.[26] M. Mursaleen, λ -statistical convergence, Math. Slovaca, 50(2000), 111-115.[27] Richard F. Patterson and E. Savas¸, Lacunary statistical convergence of doublesequences, Math. Commun., 10(1), 55-61(2005).[28] T. ˇ S al ` a t, On statistically convergent sequences of fuzzy real numbers, Math.Slovaca, 30(1980), 139-150.[29] W. H. Ruckle, FK spaces in which the sequence of coordinate vectors isbounded, Canad. J. Math., 25 (1973), 973ˆa ee