Applied Mathematics and Computation 215 (2009) 1030–1034
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On statistical limit points of double sequences Pratulananda Das a,*, Prasanta Malik a, Ekrem Savasß b a b
Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India Istanbul Commerce University, Department of Mathematics, Üsküdar-Istanbul, Turkey
a r t i c l e
i n f o
Keywords: Double sequence Double natural density Statistical convergence Statistical limit point
a b s t r a c t We investigate the structure of the set of all statistical limit points of a double sequence and prove certain results, mainly showing that this set can be characterized as a F r -set. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The notion of statistical convergence was introduced by Fast [5] and Schoenberg [16], independently. Over the years and under different names statistical convergence has been discussed in the theory of fourier analysis, ergodic theory and number theory. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy [6], Šalát [14], and Connor [2] and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its applications are also connected with subsets of the Stone–Cech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. Fridy [7] introduced the notion of a statistical limit point of a sequence x ¼ fxn gn2N of real numbers. Fridy studied the set Kx of all such points and showed that this set need not be closed or open in R. More works in this line was done by Kostyrko et al. [8]. Recently the notion of statistical convergence was introduced for double sequences by Mursaleen and Edely [11] using double natural density (also independently by Moricz [9] who studied it for multiple sequences). Since then a lot of work have been done on double sequences which can be seen from [1,3,4,10–12] also where more references can be found. Under the circumstances it seems therefore reasonable to introduce the concept of statistical limit points for double sequences. In this paper we preciously do the same and show that for double sequences the set of statistical limit points can be characterized as a F r -set in R. We also prove the statistical analogue of the well-known Van der Corput difference theorem for double sequences. As in [3,4,9–11], it again appears that the methods of proofs, though modeled after those for single sequences, are certainly not always analogous and are more complicated in nature.
2. Definitions and notations Throughout the paper N and R denote the sets of all positive integers and the set of all real numbers, respectively. As in [4] we also assume that the set N N (or any subset of N N) is ordered with respect to the relation
* Corresponding author. E-mail addresses:
[email protected] (P. Das),
[email protected] (P. Malik),
[email protected] (E. Savasß). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.036
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ði; jÞ < ði1 ; j1 Þ if i þ j < i1 þ j1 ; or i < i1
when i þ j ¼ i1 þ j1 ðAÞ
and ði; jÞ ¼ ði1 ; j1 Þ if i ¼ i1 ; j ¼ j1 : By the convergence of a double sequence we mean the convergence in Pringsheim’s sense [13]. A double sequence x ¼ fxjk gj;k2N of real numbers is said to converge to n 2 R in Pringsheim’s sense if for any > 0 there exists N 2 N such that jxjk nj < , whenever j; k P N and in this case we writelim j!1 xjk ¼ n. k!1
If A N N has the property that for any ðm; nÞ 2 N N, there is ðj; kÞ 2 A such that j > m; k > n then fxjk gðj;kÞ2A is called a subsequence of the double sequence fxjk gj;k2N . Pringsheim convergence of a subsequence is also similarly defined as above. If the subsequence fxjk gðj;kÞ2A is convergent to n 2 R then we write lim j!1 xjk ¼ n. k!1 ðj;kÞ2A
A real number a is said to be a P-limit point of a double sequence x ¼ fxjk gj;k2N if there exists a subsequence of x which is convergent to a. Let L2 ðxÞ denote the set of P-limit points of x. We now recall the concept of double natural density. Let K N N and Kðn; mÞ be the numbers of ðj; kÞ 2 K such that is called the upper double natural density of K. If the sequence j 6 n; k 6 m. Then the number d2 ðKÞ ¼ lim n!1 sup Kðn;mÞ n:m m!1 n o Kðn;mÞ has a limit in Pringsheim’s sense then we say that K has double natural density and is denoted by n:m n;m2N
d2 ðKÞ ¼ lim
n!1 m!1
Kðn; mÞ : n:m
Definition 2.1. ([11,9]). A double sequence x ¼ fxjk gj;k2N of real numbers is said to be statistically convergent to n 2 R, if for any > 0, we have d2 ðAðÞÞ ¼ 0, where AðÞ ¼ fðj; kÞ 2 N N; jxjk nj P g. The following example is giving the statistical convergence of double sequence ðxjk Þ. Example 2.1. Define the double sequence fxjk g by xjk ¼ 1 if j and k are square,=0 otherwise. Then for every
e>0
pp j k ¼ 0; d2 fðj; kÞ 2 N N : jxjk nj P eg ¼ d2 ðAðÞÞ 6 limjk jk
i.e., the set A has double natural density zero for every e > 0. This implies that st limjk jxjk nj ¼ 0 in Pringsheim’s sense. But the sequence ðxjk Þ is not convergent to n in Pringsheim’s sense. Definition 2.2. A real number a is said to be a statistical limit point of a double sequence fxjk gj;k2N if there exists a subsequence fxjk gðj;kÞ2A of fxjk gj;k2N (where A N N) such that d2 ðAÞ > 0 and lim j!1 xjk ¼ a. For a double sequence x ¼ fxjk gj;k2N we define
k!1 ðj;kÞ2A
K2 ðxÞ ¼ fa 2 R; a is a statistical limit point of xg:
Example 2.2. Let xjk ¼ 1 if k and j are square and xjk ¼ 0; otherwise: then L2 ðxÞ ¼ f0; 1g and K2 ðxÞ ¼ f0g. It is clear that K2 ðxÞ # L2 ðxÞ for any double sequence x. The following example shows that for a double sequence, we can even have K2 ðxÞ ¼ ; while L2 ðxÞ ¼ R. 2 Example 2.3. Let frjk g1 j;k¼1 be a double sequence whose range is the set of all rational numbers and define, xjk ¼ r jk , if j ¼ n and k ¼ m2 for n; m ¼ 1; 2; 3; . . . ; ¼ ðj; kÞ otherwise. Since the set of square has density zero, it follows that K2 ðxÞ ¼ ; while the fact that frjk : ðj; kÞ 2 N Ng in dense in R implies that L2 ðxÞ ¼ R:
The main object of this paper is to study the set K2 ðxÞ. 3. Topological characterization of the set 2 ðxÞ In this section we investigate the Borel classification of the set K2 ðxÞ. Theorem 3.1. For every double sequence x ¼ fxjk gj;k2N , the set K2 ðxÞ is an F r -set in R. Proof. For any 0 < t 6 1 we define
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K2 ðx; tÞ ¼ fa 2 R; 9A0 N N such that lim xjk ¼ a and d2 ðA0 Þ P tg: j!1 k!1 ðj;kÞ2A0
S 1 1 Clearly K2 ðxÞ ¼ 1 q¼1 K2 x; q . So it is sufficient to show that for each q 2 N; K2 ðx; qÞ is closed in R. Let ai 2 K2 ðx; tÞ; 0 < t 6 1 and limi!1 ai ¼ a. Then for each ai we can select a subsequence of x with index set AðiÞ such that fxjk gðj;kÞ2AðiÞ is convergent to ai and d2 ðAðiÞ Þ P t. Moreover for an arbitrary given sequence fi gi2N of positive numbers we can select N 1 < N 2 < and M 1 < M2 < such that
T ðiÞ ððNi1 ; Ni ðMi1 ; M i Þ A Ni :M i
P t i ;
S ðiÞ T for i ¼ 1; 2; 3; . . ., where for B N N; jBj denotes the cardinality of B. Now let A ¼ 1 ððN i1 ; N i ðM i1 ; M i Þ and i¼1 A limi!1 i ¼ 0. Then d2 ðAÞ P t. Now if A is ordered to increasing ðj; kÞ then lim j!1 xjk ¼ a, because k!1 ðj;kÞ2A S ðiÞ fðm; nÞ; jxmn aj < g with the exception of a set fðm; nÞ; m; n 2 Ng which is contained in a finite number of A 1 i¼j A rows and columns of N N, for any > 0 and sufficiently large j ¼ jðÞ. h
Theorem 3.2. Let X be an arbitrary F r -set in R, then there exists a double sequence x ¼ fxjk gj;k2N such that K2 ðxÞ ¼ X. S Proof. Let X ¼ 1 p¼1 X p , where X p ’s are non-empty closed sets. For every p ¼ 1; 2; 3; . . . we can select a double sequence p fxrs gr;s2N 2 X p such that the set of all P-limit points of fxprs gr;s2N coincides with X p . T S Aq ¼ / for p – q and each Ap has positive asymptotic density such that Decompose N ¼ 1 p¼1 Ap where Ap Sn d N n i¼1 Ai ! 0 as n ! 1 (cf. [7, Ex-3]). S We now define Dp ¼ Ap N. Then it is easy to check that d2 ðDp Þ ¼ dðAp Þ. Further N N ¼ 1 p¼1 Dp and Sp S1 d2 N N n i¼1 Di ! 0 as p ! 1. Again, for every p ¼ 1; 2; 3; . . . decompose Ap ¼ i¼1 C p;i into pairwise disjoint sets C p;i S having dðC p;i Þ ¼ dðAp Þ for each i ¼ 1; 2; 3; . . . (see [8]). Consequently we have Dp ¼ 1 i¼1 ðC p;i NÞ. We now rewrite the S sequence fC p;i Ngi2N as fBprs gr;s2N . Clearly we now have the decomposition Dp ¼ r;s2N Bprs into pairwise disjoint sets with d2 ðBprs Þ ¼ d2 ðDp Þ for every r; s 2 N. Next we construct a double sequence x ¼ fxjk gj;k2N in the following way. Let
xmn ¼ constant ¼ xprs forðm; nÞ 2 Bprs : Then we have the following two cases: S Case I. fxjk gðj;kÞ2A is a subsequence of x which is convergent to a and a R X. Then for each p the set pi¼1 Di contains only those ðm; nÞ which run over at most finite number of rows and columns of N N and so we must have d2 ðAÞ ¼ 0. Hence a R K2 ðxÞ. Case II. a 2 X p and 0 < < d2 ðDp Þ. By our assumption there exists a subsequence fxpjk gðj;kÞ2O (O N N) of fxpjk gj;k2N such that lim j!1 xpjk ¼ a. We write O in the increasing order (ordered by the relation (A)) as O ¼ fðj1 ; k1 Þ < ðj2 ; k2 Þ < g. k!1 ðj;kÞ2O
Now in view of the construction of the sets Bprs (each is of the form F N for some F N with dðFÞ > 0) we can select positive integers N j1 k1 < N j2 k2 < N j3 k3 < such that
n o p T Bji ki ððN ji1 ki1 ; Nji ki NÞ N ji k i Now writing B ¼
P d2 ðDp Þ :
S1 p T ððN ji1 ki1 ; N ji ki NÞ it is easy to verify that fxjk gðj;kÞ2B converges to a and d2 ðBÞ P d2 ðDp Þ > 0. i¼1 Bji ki
This implies that a 2 K2 ðxÞ. This completes the proof of the theorem.
h
Remark 1. Whenever a set of real numbers is small, a natural question arises regarding its category or more importantly porosity position. As it is known [15] that a F r -set M in a metric space Y is uniformly r-very strongly porous in Y=M so it is clear from Theorem 3.1 that for any double sequence x ¼ fxjk gj;k2N ; K2 ðxÞ is uniformly r-very strongly porous (and so a set of first category) in R=K2 ðxÞ. If x is statistically convergent (i.e. K2 ðxÞ is singleton) or if K2 ðxÞ is a finite set then clearly K2 ðxÞ is uniformly very strongly porous in R. However for a general double sequence x, the porosity behaviour of K2 ðxÞ in whole of R remains open. 4. Some further results on 2 ðxÞ In this section we prove some results on K2 ðxÞ including a result which can be thought of as a statistical analogue of wellknown Van der Corput difference theorem for double sequences. We first prove the following two basic results.
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Lemma 4.1. The function d2 ðÞ defined on PðN NÞ is sub-additive. The proof is straightforward and so is omitted. Lemma 4.2. If for A N N; d2 ðAÞ > 0 then there exists at least one p 2 N such that d2 ðA A þ ðp; pÞ ¼ fðj þ p; k þ pÞ; ðj; kÞ 2 Ag.
T ðA þ ðp; pÞÞÞ > 0 where we define
Proof. We first note that for fixed ðp; qÞ 2 N N and any ðm; nÞ 2 N N
Aðm; nÞ ¼ ðA þ ðp; qÞÞðm þ p; n þ qÞ and so
d2 ðA þ ðp; qÞÞ ¼ lim sup m!1 n!1
ðA þ ðp; qÞÞðm þ p; n þ qÞ Aðm; nÞ Aðm; nÞ m:n ¼ lim sup ¼ lim sup m!1 ðm þ pÞ:ðn þ qÞ ðm þ pÞ:ðn þ qÞ m!1 m:n ðm þ pÞ:ðn þ qÞ n!1
n!1
¼ d2 ðAÞ: Now assume that d2 ðAÞ > 0 but d2 ðA for any r; t 2 N with r < t. Because
ðj; kÞ 2 ðA þ ðr; rÞÞ
\
T T ðA þ ðp; pÞÞÞ ¼ 0 for every p 2 N. Then we claim that d2 ðððA þ ðr; rÞÞ ðA þ ðt; tÞÞÞÞ ¼ 0
ðA þ ðt; tÞÞ () ðj; kÞ ¼ ðj1 ; k1 Þ þ ðr; rÞ ¼ ðj1 þ r; k1 þ rÞ ¼ ðj2 ; k2 Þ þ ðt; tÞ ¼ ðj2 þ t; k2 þ tÞ
for some ðj1 ; k1 Þ; ðj2 ; k2 Þ 2 A
() ðj1 ; k1 Þ ¼ ðj2 ; k2 Þ þ ðt r; t rÞ \ () ðj1 ; k1 Þ 2 ðA þ ðt r; t rÞÞ A \ () ðj; kÞ 2 A ðA þ ðt r; t rÞÞ þ ðr; rÞ: T T T Hence d2 ððA þ ðr; rÞÞ ðA þ ðt; tÞÞÞ ¼ d2 ððA ðA þ ðt r; t rÞÞÞ þ ðr; rÞÞ ¼ d2 ðA ðA þ ðt r; t rÞÞÞ ¼ 0. 1 < d2 ðAÞ. Now consider the M þ 1 translations of A, namely Choose M 2 N such that M
A þ ð1; 1Þ; A þ ð2; 2Þ; A þ ð3; 3Þ; . . . ; A þ ðM þ 1; M þ 1Þ: Clearly
SMþ1 p¼1
d2
ðA þ ðp; pÞÞ N N but
Mþ1 [
! ðA þ ðp; pÞÞ
P
p¼1
Mþ1 X
d2 ðA þ ðp; pÞÞ
p¼1
Mþ1 \ X Mþ1 d2 ðA þ ðp; pÞÞ ðA þ ðq; qÞÞ ¼ d2 ðAÞ 0 > > 1; M p6Mþ1 p¼1 X
q6Mþ1
which is a contradiction. h The following theorem can be thought of as a statistical analogue of Van der Corput difference theorem for double sequences. Theorem 4.1. Let x ¼ fxjk gj;k2N be a double sequence of real numbers. If for every p ¼ 1; 2; 3; ::: the difference sequence x0 ¼ fxðjþpÞðkþpÞ xjk gj;k2N has K2 ðx0 Þ ¼ /, then K2 ðxÞ ¼ /. Assume that K2 ðxÞ–/. Then there exists a a 2 R and a subsequence fxjk gðj;kÞ2A (A N N) such that lim j!1 xjk ¼ a and
d2 ðAÞ > 0.
By
lim j!1 k ! 1T ðj þ p; k þ pÞ 2 ðA A þ ðp; pÞÞ
Lemma
4.2,
there
exists
a
p2N
such
that
d2 ðA
T
k!1 ðj;kÞ2A
ðA þ ðp; pÞÞÞ > 0.
Then
clearly
ðxðjþpÞðkþpÞ xjk Þ ¼ 0.
In Theorem 3.2 we had constructed for an arbitrary F r -set X, a double sequence x ¼ fxjk gj;k2N such that K2 ðxÞ ¼ X. A natural question can arise whether this double sequence x is unique. The following theorem provides an answer in the negative. Theorem 4.2. For any two double sequences x ¼ fxjk gj;k2N and y ¼ fyjk gj;k2N we have
lim
M!1 N!1
1 X jxjk yjk j ¼ 0 ) K2 ðxÞ ¼ K2 ðyÞ: M:N 16j6M 16k6N
Proof. We claim that, for any
> 0,
d2 ðfðj; kÞ 2 N N; jxjk yjk j P gÞ ¼ 0: For otherwise if there exists an
o > 0 for which d2 ðfðj; kÞ 2 N N; jxjk yjk j P o gÞ ¼ aðsayÞ > 0 then it is easy to show that
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lim sup
M!1 N!1
1 X jxjk yjk j P a o > 0; M:N 16j6M 16k6N
which is a contradiction to our assumption. Thus fjxjk yjk jgj;k2N is statistically convergent to 0 and so by Theorem 2.1 [11], there exists D N N with d2 ðDÞ ¼ 1 and lim j!1 jxjk yjk j ¼ 0. Now let b 2 K2 ðxÞ. Then there is a subsequence fxjk gðj;kÞ2A k!1 ðj;kÞ2D
(A N N) of x such that d2 ðAÞ > 0 and lim j!1 xjk ¼ b. Clearly then d2 ðA k!1 ðj;kÞ2A
T
DÞ ¼ d2 ðAÞ > 0 and lim j!1 yjk ¼ b. Therefore k!1 ðj;kÞ2A
b 2 K2 ðyÞ. This proves the theorem. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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