In this paper we extend concepts of statistical limit superior and
inferior (as introduced by Fridy and Orhan) to $mathcal{I-}$limit superior and inferior and give some $mathcal{I-}$analogue of properties of statistical limit superior and inferior for a sequence of real numbers. Also we extend the concept of the statistical core to $mathcal{I-}$core for a complex number sequence and get necessary conditions for a summability
matrix $A$ to yield $mathcal{I-}coreleft{ Axright} $ $subseteq 
mathcal{I-}coreleft{ xright} $ whenever $x$ is a bounded complex number
sequence

K. Demirci

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Mathematical Communications 6(2001), 165-172 165I−limit superior and limit inferiorKamil Demirci∗Abstract. In this paper we extend concepts of statistical limitsuperior and inferior (as introduced by Fridy and Orhan) to I−limitsuperior and inferior and give some I−analogue of properties of statis-tical limit superior and inferior for a sequence of real numbers. Also weextend the concept of the statistical core to I−core for a complex num-ber sequence and get necessary conditions for a summability matrix Ato yield I−core {Ax} ⊆ I−core {x} whenever x is a bounded complexnumber sequence.Key words: statistical limit superior and inferior, statistical coreof a sequence, I−convergent sequenceAMS subject classifications: Primary 40A05; Secondary26A03,11B05Received November 2, 2001 Accepted December 20, 20011. IntroductionIf K is a subset of natural numbers N, Kn will denote the set {k ∈ K : k ≤ n}and |Kn| will denote the cardinality of Kn. Natural density of K [20], [13] isgiven by δ(K) := limn 1n |Kn| , if it exists. Fast introduced the definition of astatistical convergence using the natural density of a set. The number sequencex = (xk) is statistically convergent to L provided that for every ε > 0 the setK := K(ε) := {k ∈ N : |xk − L| ≥ ε} has natural density zero [7] ,[9] . Hence xis statistically convergent to L iff (C1χK(ε))n → 0, (as n → ∞, for ever ε > 0 ),where C1 is the Cesa´ro mean of order one and χK is the characteristic functionof the set K. Properties of statistically convergent sequences have been studied in[1] , [2] , [9] , [18] , [21].Statistical convergence can be generalized by using a nonnegative regular sum-mability matrix A in place of C1.Following Freedman and Sember [8] , we say that a set K ⊆ N has A−density ifδA(K) := limn(AχK)n = limn∑k∈K ank exists where A = (ank) is a nonnegativeregular matrix.The number sequence x = (xk) is A−statistically convergent to L provided thatfor every ε > 0 the set K(ε) has A−density zero[2], [8] , [18].∗Department of Mathematics, Faculty of Sciences and Arts, Kırıkkale University, Yahs¸ihan71450, Kırıkkale, Turkey166 K.DemirciFridy [10] has introduced the notions of a statistical limit point and a clusterpoint. Fridy and Orhan [11] studied the idea of statistical limit superior and inferior.Connor and Kline [4] and Demirci [6] extended these concepts to A−statisticalconvergence using a nonnegative regular summability matrix A in place of C1. AlsoConnor has introduced µ−statistical analogue of these concepts using a finitelyadditive set function µ taking values in [0, 1] defined on a field Γ of subsets of Nsuch that if |A| <∞ , then ; if A ⊂ B and µ(B) = 0 , then µ(A) = 0 ; and µ(N) = 1[3] , [5] .The number sequence x = (xk) is µ−statistically convergent to L provided thatµ({k ∈ N : |xk − L| ≥ ε}) = 0 for every ε > 0 [3], [5].Kostyrko, Macˇaj and Sˇala´t [15] , [16] introduced the concepts of I−convergence,I−limit point and I−cluster point of sequences of real numbers based on the notionof the ideal of subsets of N.In this paper we extend concepts of statistical limit superior and inferior toI−limit superior and inferior and give some properties of I−limit superior andinferior for a sequence of real numbers. We also extend the concept of a statisticalcore to I−core for a complex number sequence and get necessary conditions for asummability matrix A to yield I−core {Ax} ⊆ I−core {x} whenever x is a boundedcomplex number sequence.2. Definition and notationsWe first recall the concepts of an ideal and a filter of sets.Definition 1. Let X = φ. A class S ⊆ 2X of subsets of X is said to be an idealin X provided that S is additive and hereditary , i.e. if S satisfies the conditions:(i) φ ∈ S,(ii) A,B ∈ S ⇒ A ∪B ∈ S,(iii) A ∈ S, B ⊆ A ⇒ B ∈ S([17] ,p.34).An ideal is called non-trivial if X /∈ S.Definition 2. Let X = φ. A non-empty class  ⊆ 2X of subsets of X is saidto be a filter in X provided that:(i) φ ∈ ,(ii) A,B ∈ ⇒ A ∩B ∈ ,(iii) φ ∈ , A ⊆ B ⇒ B ∈ ([19] ,p.44).The following proposition expresses a relation between the notions of an idealand a filter:Proposition 1. Let S be non-trivial in X, X = φ. Then the class(S) = {M ⊆ X : ∃A ∈ S :M = X \A}is a filter on X (we will call (S) the filter associated with S ) [15] .Definition 3. A non-trivial ideal S in X is called admissible if {x} ∈ S foreach x ∈ X [15].As usual, R will denote real numbers and C complex numbers.I−limit superior and limit inferior 167Definition 4. Let I be a non-trivial ideal in N. Then(i) A sequence x = (xn) of real numbers is said to be I−convergent to L ∈ R iffor every ε > 0 the set A(ε) = {n : |xn − L| ≥ ε} belongs to I [15] . In this casewe write I− limx = L.(ii) An element ξ ∈ R is said to be I−limit point of the real number sequencex = (xn) provided that there exists a set M = {m1 < m2 < ...} ⊂ N such that M /∈I and limk xmk = ξ [16] .(iii) An element ξ ∈ R is said to be I−cluster point of the real number sequencex = (xn) iff for each ε > 0 we have {k : |xk − ξ| < ε} /∈ I [16] .Note that the set of I−cluster points of x is a closed points set in R where I isan admissible ideal [15] .Some results on I−convergence, I−limit point and I−cluster point may befound in [15],[16].Throughout the paper I will be an admissible ideal.3. I−limit superior and inferiorIn this section we study the concepts of I−limit superior and inferior for a realnumber sequence.For a real number sequence x = (xk) let Bx denote the setBx := {b ∈ R : {k : xk > b} /∈ I} .Similarly,Ax := {a ∈ R : {k : xk < a} /∈ I} .We begin with a definition.Definition 5. Let I be an admissible ideal and x a real number sequence. Thenthe I−limit superior of x is given byI− lim supx :={supBx, if Bx = φ,−∞, if Bx = φ.Also, the I−limit inferior of x is given byI− lim inf x :={inf Ax, if Ax = φ,+∞, if Ax = φ.Note that if we define I = {K ⊆ N : δA(K) = 0} , I = {K ⊆ N : δ(K) = 0}and I = {K ∈ Γ : µ(K) = 0} in Definition 5, then we get Definition 1 of [6], Def-inition 1 of [11] and Connor’s definitions [5] of µ−statistical superior and inferior,respectively. This observation suggests the following result which can be proved bya straightforward least upper bound argument.Theorem 1. If β = I− lim supx is finite, then for every positive number ε{k : xk > β − ε} /∈ I and {k : xk > β + ε} ∈ I. (1)Conversely, if (1) holds for every positive ε, then β = I− lim supx.The dual statement for I− lim inf x is as follows.168 K.DemirciTheorem 2. If α = I− lim inf x is finite, then for every positive ε{k : xk < α+ ε} /∈ I and {k : xk < α− ε} ∈ I. (2)Conversely, if (2) holds for every positive ε, then α = I− lim inf x.Considering the definition of I−cluster point in Definition 4 we see that The-orems 1 and 2 can be interpreted as saying that I− lim supx and I− lim inf x arethe greatest and the least I−cluster points of x.Now we have the followingTheorem 3. For any real number sequence x,I− lim inf x ≤ I− lim supx.Proof. First consider the case in which I− lim supx = −∞. Hence we haveBx = φ, so for every b in R , {k : xk > b} ∈ I which implies that {k : xk ≤ b} ∈(I) so for every a in R, {k : xk ≤ a} /∈ I . Hence I− lim inf x = −∞.The case in which I− lim supx = +∞ needs no proof, so we next assume thatβ = I− lim supx is finite, and α := I− lim inf x. Given ε > 0 we show thatβ + ε ∈ Ax, so that α ≤ β + ε. By Theorem 1, {k : xk > β + ε} ∈ I becauseβ = lub Bx. This implies{k : xk ≤ β + ε2} ∈ (I). Since {k : xk ≤ β + ε2} ⊆{k : xk < β + ε} and (I) is a filter on N, {k : xk < β + ε} ∈ (I). This implies{k : xk < β + ε} /∈ I. Hence β+ε ∈ Ax. By definition α = inf Ax, so we concludethat α ≤ β + ε ; and since ε is arbitrary this proves that α ≤ β. ✷From Theorem 3 and Definition 5, it is clear thatlim inf x ≤ I− lim inf x ≤ I− lim supx ≤ lim supxfor any real number sequence x.I−limit point of a sequence x is defined in (ii) of Definition 4 as the limit of asubsequence of x whose indices do not belong to I. We cannot say that I− lim supxis equal to the greatest I−limit points of x. This can be seen from Example 4 in[11] where I = {K ⊆ N : δ(K) = 0}.Definition 6. The real number sequence x = (xk) is said to be I−bounded ifthere is a number B such that {k : |xk| > B} ∈ I.Note that I−boundedness implies that I− lim sup and I− lim inf are finite, soproperties (1) and (2) of Theorems 1and 2 hold.Theorem 4. The I−bounded sequence x is I−convergent if and only ifI− lim inf x = I− lim supx.Proof. Let α := I− lim inf x and β := I− lim supx. First suppose that I− limx = L and ε > 0. Then {k : |xk − L| ≥ ε} ∈ I, so {k : xk > L+ ε} ∈ I, whichimplies that β ≤ L. We also have {k : xk < L− ε} ∈ I, which yields that L ≤ α.Therefore β ≤ α. Combining this with Theorem 3 we conclude that α = β.Now assume α = β and define L := α. If ε > 0 then (1) and (2) of Theorem 1and 2 imply{k : xk > L+ ε2} ∈ I and {k : xk < L− ε2} ∈ I. Hence I− limx = L. ✷I−limit superior and limit inferior 1694. I− coreIn [11] Fridy and Orhan introduced the concept of the statistical core of a realnumber sequence, and proved the statistical core theorem. Those results have alsobeen extended to the complex case too [12] . Using the same technique as in [12], weintroduce the concept of I−core of a complex sequence and get necessary conditionsfor a summability matrix A to yield I−core {Ax} ⊆ I−core {x} whenever x is abounded complex number sequence.In this section x, y and z will denote complex number sequences and A =(ank) will denote an infinite matrix of complex entries which transforms a complexnumber sequence x = (xk) into the sequence Ax whose n-th term is given by(Ax)n =∑∞k=1 ankxk.In [14] the Knopp core of the sequence x is defined byK−core {x} := ∩n∈NCn(x),where Cn(x) is the closed convex hull of {xk}k≥n. In [22] it is shown that for everybounded xK−core {x} := ∩z∈CB∗x(z),where B∗x(z) := {w ∈ C : |w − z| ≤ lim supk |xk − z| } .The next definition is an I−analogue of statistical core [12] of a sequence.Note that, if x and y are sequences such that {k ∈ N : xk = yk} /∈ I,then wewrite “xk = yk, for I− a.a. k”.Definition 7. Let I be an admissible ideal in N. For any complex sequence xlet HI(x) be the collection of all closed half-planes that contain xk for I− a.a. k;i.e.,HI(x) := {H : is a closed half-plane {k ∈ N : xk /∈ H} ∈ I} ,then the I−core of x is given byI−core {x} := ∩H∈HI(x)H.It is clear that I−core {x} ⊆ K−core {x} for all x. AlsoI−core {x} = [I− lim inf x , I− lim supx]for any I−bounded real number sequence.The next theorem is an I−analogue of the Lemma of [12].Theorem 5. Let I be an admissible ideal in N and assume that x is anI−bounded sequence; for each z ∈ C letBx(z) :={w ∈ C : |w − z| ≤ I− lim supk|xk − z|};then I−core {x} := ∩z∈CBx(z).Proof. From the definition of I− lim supx and Theorem 1, observe that the diskBx(z) is equal to the intersection of all closed disks centered at z that contain xk170 K.Demircifor I− a.a.k. First assume w /∈ ∩z∈CBx(z), say w /∈ ∩z∈CBx(z∗) for some z∗. LetH be the half-plane containing Bx(z∗) whose boundary line is perpendicular to theline containing w and z∗ and tangent to the circular boundary of Bx(z∗). SinceBx(z∗) ⊂ H and Bx(z∗) contains xk for I− a.a.k, it follows that H ∈ HI(x). Sincew /∈ H , this implies w /∈ ∩H∈HI (x)H . Hence, I−core{x} ⊆ ∩z∈CBx(z).Conversely, w /∈ ∩H∈HI (x)H , let H be a plane in HI(x) such that w /∈ H . Letbe the line through w that is perpendicular to the boundary of H and let p be themid-point of the segment to L between w and H . Let z be a point of L such thatz ∈ H and consider the diskB(z) := {ξ ∈ C : |ξ − z| ≤ |p− z| } .Since x is I−bounded and xk ∈ H I− a.a.k, we can choose z sufficiently farfrom p so that |p− z| = I− lim supk |xk − z|. Thus B(z) is one of the Bx(z) disks,and since w /∈ B(z), we get that w /∈ ∩z∈CBx(z). This establishes the proof. ✷We note that Theorem 5 is not necessarily valid if x is not I−bounded. Thiscan be seen from Remark in [12] where I = {K ⊆ N : δ(K) = 0}.Throughout the remainder of this paper the set of bounded complex sequenceswill be denoted by *∞.Now we give necessary conditions on matrix A so that the Knopp core of Ax iscontained in the I−core of x for every bounded complex number sequence.Theorem 6. Let I be an admissible ideal in N. If matrix A satisfiessupn∑∞k=1 |ank| <∞ and the following conditions(i) A regular and limn∑k∈E |ank| = 0 whenever E ∈ I;(ii) limn∑∞k=1 |ank| = 1,then K−core {Ax} ⊆ I−core {x} for every x ∈ *∞.Proof. Assume (i) and (ii) and let w ∈ K−core {Ax}. For any z ∈ C we have|w − z| ≤ lim supn|z − (Ax)n|= lim supn∣∣∣∣∣z −∞∑k=1ankxk∣∣∣∣∣= lim supn∣∣∣∣∣∞∑k=1ank (z − xk) + z (1 −∞∑k=1ank)∣∣∣∣∣≤ lim supn∣∣∣∣∣∞∑k=1ank (z − xk)∣∣∣∣∣+ lim supn |z|∣∣∣∣∣1−∞∑k=1ank∣∣∣∣∣= lim supn∣∣∣∣∣∞∑k=1ank (z − xk)∣∣∣∣∣ . (3)Let r = I− lim supn |xn − z|, suppose ε > 0, and letE := {k : |zk − L| > r + ε}.Then E ∈ I, and we haveI−limit superior and limit inferior 171∣∣∣∣∣∞∑k=1ank (z − xk)∣∣∣∣∣ =∣∣∣∣∣∑k∈Eank (z − xk) +∑k/∈Eank (z − xk)∣∣∣∣∣≤∑k∈E|ank| |z − xk|+∑k/∈E|ank| |z − xk|≤ supk|z − xk|∑k∈E|ank|+ (r + ε)∑k/∈E|ank| .Now (i) and (ii) imply thatlim supn=∣∣∣∣∣∞∑k=1ank (z − xk)∣∣∣∣∣ ≤ r + ε.It follows from (3) that |w − z| ≤ r + ε; and since ε is arbitrary, this yields|w − z| ≤ r. Hence,w /∈ Bx(z) so by the Theorem 5 we get w ∈ I−core {x}. Hencethe proof is completed. ✷Since I−core {x} ⊆ K−core {x}, we have the following corollary.Corollary 1. If matrix A satisfies supn∑∞k=1 |ank| <∞ and properties (i) and(ii) of Theorem6, thenI−core {Ax} ⊆ I−core {x}for every x ∈ *∞.Note that the converse of Corollary 1 does not hold. This can be seen fromExample in [12] where I = {K ⊆ N : δ(K) = 0}.References[1] J. Connor, The statistical and strong p−Cesa´ro convergence of sequences,Analysis 8(1988), 47-63.[2] J. Connor, On strong matrix summability with respect to a modulus and sta-tistical convergence, Canad. Math. Bull. 32(1989), 194-198.[3] J. Connor, Two valued measures and summability, Analysis 10(1990), 373-385.[4] J. Connor, J.Kline, On statistical limit points and the consistency of statis-tical convergence, J. Math. Anal. Appl. 197(1996), 392-389.[5] J. Connor, A topological and functional analytic approach to statistical con-vergence, In: Analysis of Divergence, Birkhauser, Boston, 1999, 403-413.[6] K.Demirci, A−statistical core of a sequence, Demonstratio Math. 33(2000),343-353.[7] H.Fast, Sur la convergence statistique, Colloq. Math. 2(1951), 241-244.172 K.Demirci[8] A.R. Freedman, J. J. Sember, Densities and summability, Pacific J. Math.95(1981), 293-305.[9] J.A. Fridy, On statistical convergence, Analysis 5(1985), 301-313.[10] J.A. Fridy, Statistical limit points, Proc. Amer. Math. Soc. 118(1993), 1187-1192.[11] J.A. Fridy, C.Orhan, Statistical limit superior and limit inferior, Proc.Amer. Math. Soc. 125(1997), 3 625-3 631.[12] J.A. Fridy, C.Orhan, Statistical core theorems, J. Math. Anal. Appl.208(1997), 520-527.[13] H.Halberstem, K. F.Roth, Sequences, Springer-Verlag, New York, 1993.[14] H.Hardy, Divergent Series, Oxford Univ. Press, London, 1949.[15] P.Kostyrko, M.Macˇaj, T. Sˇala´t, Statistical convergence andI−convergence, to appear in Real Anal. Exchange.[16] P.Kostyrko, M.Macˇaj, T. Sˇala´t, I− convergence, to appear in RealAnal. Exchange.[17] C.Kuratowski, Topologie I, PWN, Warszava, 1958.[18] H. I.Miller, A measure theoretical subsequence characterization of statisticalconvergence, Trans. Amer. Math. Soc. 347(1995), 1811-1819.[19] J.Nagata, Modern General Topology, North-Holland Publ. Comp.,Amsterdam-London, 1974.[20] I.Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers, 4thEd., John Wiley and Sons, New York, 1980.[21] T. Sˇala´t, On statistically convergent sequences of real numbers, Math. Slovaca30(1980), 139-150.[22] A.A. Shcherbakov, Kernels of sequences of complex numbers and their re-gular transformations, Math. Notes 22(1977), 948-953.

I-limit superior and limit inferior

In this paper we extend concepts of statistical limit superior and inferior (as introduced by Fridy and Orhan) to I−limit superior and inferior and give some I−analogue of properties of statistical limit superior and inferior for a sequence of real numbers. Also we extend the concept of the statistical core to I−core for a complex number sequence and get necessary conditions for a summability matrix A to yield I−core {Ax} ⊆ I−core {x} whenever x is a bounded complex number sequence

I-limit superior and limit inferior

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