ON SOME PROPERTIES OF I-CONVERGENCE ˇ AT, ´ ˇ ZIMAN TIBOR SAL BINOD CHANDRA TRIPATHY AND MILOS
Abstract. In the paper [5] the concept of I-convergence is introduced. This concept is a generalization of the statistical convergence. In this paper some notions and results from the statistical convergence are extended to the Iconvergence.
Introduction The notion of a statistical convergence was introduced by Fast [1] and Schoenberg [6] independently. Later on it was further investigated from a sequence space point ˇ at [7], Tripathy of view and linked with the Summability Theory by Fridy [3], Sal´ [9, 10] and many others. The notion of an I-convergence is a generalization of the statistical convergence. ˇ at and Wilczy´ It was introduced by Kostyrko, Sal´ nski [5]. Some further results connected with the notion of the I-convergence can be found in [4] and [8]. In this papers we extend some notions and results known for the statistical convergence to the I-convergence. 1. Definitions and preliminaries Throughout the article w, c, c0 , `∞ denote the spaces of all, convergent, null and bounded complex valued sequences, respectively. Definition 1.1. A non-void class I ⊆ 2N is called an ideal if I is additive (i.e., A, B ∈ I ⇒ A ∪ B ∈ I) and hereditary (i.e., A ∈ I & B ⊆ A ⇒ B ∈ I). An ideal I is said to be non-trivial if I 6= 2N . A non-trivial ideal I is said to be admissible if I contains every finite subset of N. All ideals in this paper are assumed to be admissible. For any ideal there is a filter F(I) corresponding with I, given by F(I) = {K ⊆ N : N r K ∈ I}. Definition 1.2. A sequence x = (xn ) ∈ w is said to be I-convergent if there exists L ∈ C such that for all ε > 0, the set {n ∈ N : |xn − L| ≥ ε} ∈ I. We say, x I-converges to L, and write I- lim x = L. The number L is called I-limit of x (cf. [5]). The usual convergence is a special case of I-convergence (I = I f - the ideal of all finite subsets of N). The statistical convergence o n (see [1] or [6]) is also the special = 0 , case of I-convergence. In this case I = I d = A ⊆ N : lim |A∩{1,2,...,n}| n n→∞
2000 Mathematics Subject Classification. 40A05. Key words and phrases. density, solid space, I-convergence, symmetric space. 1
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ˇ ´ ˇ ZIMAN TIBOR SAL AT, BINOD CHANDRA TRIPATHY AND MILOS
|M | being the cardinality of the set M . For other examples of I-convergence see [5]. Remark. One can readily see that the I-convergence of sequences of points in a metric space can be defined analogously. The class of all I-convergent sequences is denoted by cI and I-null sequences (i.e, I- lim x = 0) by cI0 . Definition 1.3. A sequence (xn ) ∈ w is said to be: • I*-convergent to L (I ∗ - lim xn = L) if there is a set {n1 < n2 < · · · } ∈ n→∞
F(I) such that lim xni = L; i→∞
• I-bounded if there exists an M > 0 such that {k ∈ N : |xk | > M } ∈ I. 2. Symmetry and monotonicity of cI Definition 2.1. A set E ⊆ w is said to be: • solid if (αn xn ) ∈ E whenever (xn ) ∈ E, (αn ) ∈ w and |αn | ≤ 1 for every n ∈ N; • symmetric if (xπ(n) ) ∈ E whenever (xn ) ∈ E and π is a permutation of N. Definition 2.2. Let K = {k1 < k2 < . . . } ⊆ N and E be a sequence space. A K-step space of E is a sequence space λE K = {(xkn ) ∈ w : (xn ) ∈ E}. A canonical preimage of a sequence (xkn ) ∈ λE K is a sequence (yn ) ∈ w defined as follows: ( xn , if n ∈ K; (1) yn = 0, otherwise. A canonical preimage of a step space λE K is a set of canonical preimages of all elements in λE , i.e., y is in canonical preimage of λE K K if and only if y is canonical E preimage of some x ∈ λK . Definition 2.3. A sequence space E is said to be monotone if it contains the canonical preimage of all its step spaces (i.e., if for all infinite K ⊆ N and (xn ) ∈ E the sequence (αn xn ), where αn = 1 for n ∈ K and αn = 0 otherwise, belongs to E). The following proposition is obvious. Proposition 2.4. Every solid sequence space is monotone. The following lemmas will be used for establishing some results of the present article. ˇ at and Wilczy´ Lemma 2.5 (Kostyrko, Sal´ nski [5], Proposition 3.2). Let I be an ∗ admissible ideal. If I - lim x = L then I- lim x = L. ˇ at, Tripathy and Ziman [8], Lemma 2.5). Let K ∈ F(I) and Lemma 2.6 (Sal´ M ⊆ N. If M ∈ / I then M ∩ K ∈ / I. Theorem 2.7. If I is not maximal, then the space cI is neither solid nor monotone.
ON SOME PROPERTIES OF I-CONVERGENCE
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Proof. By Proposition 2.4 it is enough to show that cI is not monotone. Let xn = 1 for all n ∈ N. Then obviously (xn ) ∈ cI . And let K ⊆ N be such that K ∈ / I and N r K ∈ / I. Define the sequence (yn ) by (1). Then (yn ) belongs to canonical preimage of K-step space of cI , but (yn ) ∈ / cI . Hence cI is not monotone. Remark. By [8, Proposition 4.3] cI is solid for every maximal ideal I. Remark. By [8, Proposition 3.9] cI0 is solid for every non-maximal ideal. The case of maximal ideal can be treated similarly as in [8, Proposition 4.3]. Theorem 2.8. If I is not maximal and I 6= I f , then cI and cI0 are not symmetric. Proof. Let A ∈ I be infinite. If ( 1, for n ∈ A; xn = 0, otherwise, then by Lemma 2.5(xn ) ∈ cI0 ⊆ cI . Let K ⊆ N be such that K ∈ / I and N r K ∈ / I. Let ϕ : K → A and ψ : N r K → N r A be bijections. (All four sets are infinite.) Then the map π : N → N defined by: ( ϕ(n), if n ∈ K; π(n) = ψ(n), otherwise is a permutation of N, but (xπ(n) ) ∈ / cI and (xπ(n) ) ∈ / cI0 . Hence cI0 and cI are not symmetric. 3. Real sequences During this section we restrict our attention to real valued sequences. We give some simple but useful remarks about properties of these sequences. Definition 3.1. A real valued sequence x = (xn ) is said to be I-monotonic increasing (I-monotonic decreasing), if there is a set {k1 < k2 < · · · } ∈ F(I) such that xki ≤ xki+1 (xki ≥ xki+1 ) for every i ∈ N; Proposition 3.2. A real sequence x = (xn ) is I-monotonic increasing (I-monotonic decreasing) if and only if x can be written in the form (2)
x = y + z;
where y = (yn ) is nondecreasing (nonincreasing) real sequence and z = (zn ) is such that {n ∈ N; zn 6= 0} ∈ I. in this case, if y is bounded, then x is I-convergent and I- lim x = limn→∞ yn . Proof. Assume that x is I-monotonic increasing. The other case can be shoved analogously. Then there exists a set K = {k1 < k2 < · · · } ∈ F(I) such that xki ≤ xki+1 for every i ∈ N. Denote mn = max{k ∈ K; k ≤ n}. Now we can define the sequences (yn ), (zn ) as follows: (3) (4)
yn = xmn , ( 0, zn = xn ,
n∈N n ∈ K; otherwise.
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ˇ ´ ˇ ZIMAN TIBOR SAL AT, BINOD CHANDRA TRIPATHY AND MILOS
Clearly, x = y + z, y is nondecreasing and {n ∈ N; zn 6= 0} ∈ I. The converse implication is obvious. If x = y + z, y, z with described properties, and y is bounded, then it is convergent. It is a routine work to verify that x is I-convergent and I- lim x = limn→∞ yn . Corrolary 3.3. An I-monotonic sequence is I-convergent if and only if it is Ibounded. Proof. Let x be I-monotonic increasing and I-convergent sequence. Let x = y + z the decomposition of x from previous proposition. It is clear that I- lim z = 0. Hence, as y = x − z, y I-convergent and I- lim y = I- lim x. Then, it follows from monotonicity of y that y is convergent and so bounded. Thus x is I-bounded. If x is I-bounded, then the decomposition x = y + z can be made such that y is bounded: Let K = {k1 < k2 < · · · } ∈ F(I) such that xki ≤ xki+1 for every i ∈ N and M > 0 be such that K 0 = {n ∈ N |xn | ≤ M } ∈ F(I). Denote mn = max{k ∈ K ∩ K 0 ; k ≤ n} and define the sequences (yn ), (zn ) by (3) and (4). The rest follows from previous proposition. Proposition 3.4. Let x = (xn ), y = (yn ) and z = (zn ) be real sequences such that xn ≤ yn ≤ zn for all n ∈ K, where K ∈ F(I). If I- lim x = L = I- lim z then I- lim y = L. Proof. For a given ε > 0, let Bx = {n ∈ N ; |xn − L| < ε}, By = {n ∈ N ; |yn − L| < ε}, Bz = {n ∈ N ; |zn − L| < ε}. Then Bx ∩ Bz ∩ K ⊆ By , and hence By ∈ F(I). It follows that I- lim y = L.
4. Convergence field of I-convergence and I-limit of a sequence In this section we will study some properties of the function, which assigns to every I-convergent sequence its I-limit. Definition 4.1. A convergence field of I-convergence is a set F (I) = {x = (xn ) ∈ `∞ : there exists I- lim x ∈ R}. Remark. The convergence field F (I) is a closed linear subspace of `∞ with respect to sup-norm, F (I) = `∞ ∩ cI (cf. [5]). Define a function g : F (I) → R: g(x) = I- lim x,
x ∈ F (I).
Theorem 4.2. The function g : F (I) → R is Lipschitz function and hence uniformly continuous. Proof. Let x, y ∈ F (I), x 6= y. Then the sets Ax = {n ∈ N : |xn − g(x)| ≥ kx − yk}, Ay = {n ∈ N : |yn − g(y)| ≥ kx − yk}
ON SOME PROPERTIES OF I-CONVERGENCE
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belong to I. Thus the sets Bx = N r Ax = {n ∈ N : |xn − g(x)| < kx − yk}, By = N r Ay = {n ∈ N : |yn − g(y)| < kx − yk} belong to F(I). Hence also B = Bx ∩ By ∈ F(I), so that B 6= ∅. Now, taking n ∈ B, |g(x) − g(y)| ≤ |g(x) − xn | + |xn − yn | + |yn − g(y)| ≤ 3kx − yk. This means that g is Lipschitz function.
Proposition 4.3. If x, y ∈ F (I) then xy ∈ F (I) and g(xy) = g(x)g(y). Proof. Let ε > 0. The sets Bx = {n ∈ N : |xn − g(x)| < ε}, By = {n ∈ N : |yn − g(y)| < ε} belong to F(I). Now, (5)
|xn yn − g(x)g(y)| = |xn yn − xn g(y) + xn g(y) − g(x)g(y)| ≤ |xn ||yn − g(y)| + |g(y)||xn − g(x)|.
As F (I) ⊆ `∞ , there exists an M ∈ R such that |xn | < M and |g(y)| < M . Thus, using (5), we get |xn yn − g(x)g(y)| ≤ M ε + M ε = 2M ε for all n ∈ Bx ∩ By (∈ F(I)). Hence xy ∈ F (I) and g(xy) = g(x)g(y).
5. Cauchy condition for I-convergence In the case I = I d (statistical convergence) J. A. Fridy [2] proved the corresponding Cauchy condition for (statistical) convergence of a sequence. We will generalize this result for I-convergence: Theorem 5.1. A sequence x = (xn ) ∈ w I-converges if and only if for every ε > 0 there exists Nε ∈ N such that (6)
{n ∈ N : |xn − xNε | < ε} ∈ F(I).
Proof. Let L = I- lim x. Then Bε = {n ∈ N : |xn − L| < 2ε } ∈ F(I) for all ε > 0. Fix an Nε ∈ Bε . Then the following holds for all n ∈ Bε : ε ε |xNε − xn | ≤ |xNε − L| + |L − xn | < + = ε. 2 2 Hence (6) holds. Now, let (6) hold for all ε > 0. Then the set Cε = {n ∈ N : xn ∈ [xNε − ε, xNε + ε]} ∈ F(I) for all ε > 0. Denote Jε = [xNε − ε, xNε + ε]. Fix an ε > 0. Then Cε ∈ F(I) and C 2ε ∈ F(I). Hence Cε ∩ C 2ε ∈ F(I). This implies J = Jε ∩ J 2ε 6= ∅, {n ∈ N : xn ∈ J} ∈ F(I), 1 diam(J) ≤ diam(Jε ). 2
ˇ ´ ˇ ZIMAN TIBOR SAL AT, BINOD CHANDRA TRIPATHY AND MILOS
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(diam(I) denotes the length of the interval I.) This way, by induction, we can construct the sequence of (closed) intervals Jε = I0 ⊇ I1 ⊇ · · · ⊇ In ⊇ · · · with the property diam(In ) ≤ 21 diam(In−1 ) (n =T 2, 3, . . . ) and {k ∈ N : xk ∈ In } ∈ F(I) (n = 1, 2, . . . ). Then there exists a ξ ∈ In and it is a routine work to n∈N
verify that ξ = I- lim x.
References
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Fast, H.: Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. Fridy, J. A.: On statistical convergence, Analysis 5 (1985), 301–313. Fridy, J. A.: Statistical limit points, Proc. Amer. Math. Soc. 11 (4) (1993), 1187-1192. ˇ at, T.: I-convergence and extremal I-limit Kostyrko, P., Maˇ caj, M., Sleziak, M. and Sal´ points, (to appear). ˇ at, T. and Wilczy´ Kostyrko, P., Sal´ nski, W.: I-convergence, Real Anal. Exchange 26 (2000/2001), 669–686. Schoenberg, I. J.: The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375. ˇ at, T.: On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), Sal´ 139–150. ˇ at, T., Tripathy, B. C., Ziman, M.: On I-convergence field, to appear in Italian J. Pure Sal´ Appl. Math. 19/20. Tripathy, B. C.: Matrix transformation between some classes of sequences, J. Math. Analysis and Appl. 206 (2) (1997), 448-450. Tripathy, B. C.: On statistical convergence, Proc. Estonian Acad. Sci. Phys. Math. 47 (4) (1998), 299–303.
´ doFaculty of Mathematics, Physics and Informatics, Comenius University, Mlynska lina, 842 48 Bratislava, SLOVAKIA Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Khanapara, Guwahati - 718 022, INDIA E-mail address:
[email protected] ´ doFaculty of Mathematics, Physics and Informatics, Comenius University, Mlynska lina, 842 48 Bratislava, SLOVAKIA E-mail address:
[email protected]
