Selçuk J. Appl. Math. Selçuk Journal of Vol. 8. No.2. pp. 49 - 55 , 2007 Applied Mathematics
Triple Sequences and Their Statistical Convergence Ahmet ¸Sahiner, Mehmet Gürdal and F. Kadriye Düden
Department of Mathematics, Suleyman Demirel University, 32260, Isparta, Turkey e-mail:sahiner@ fef.sdu.edu.tr,gurdal@ fef.sdu.edu.tr,fatm akadriye@ hotm ail.com
Received : July 25, 2007
Summary. Pringsheim gave the definition of the convergence for double se-quences in 1900 The idea of statistical convergence was first presented by Fast in 1951 In 2003 Tripathy and Mursaleen et.al introduced statistical convergence of double sequences. In this article we introduce the notions of triple sequences and statistically convergent triple sequence. We also introduce the notions of statistical Cauchy for triple sequences, a necessary and sufficient condition for a triple statistical Cauchy sequence to be a convergent triple sequence is given. Key words: Double sequence; triple sequence; statistical convergence; triple natural density
[2000]Primary 40A05; Secondary 26A03 1. Introduction
Pringsheim gave the definition of the convergence for double sequences in 1900. Since then, this concept has been studied by many authors, see for instance [4, 7, 9, 10] The idea of statistical convergence was first presented by Fast in 1951 and Schoenberg in 1959 independently. Later on by the studies of Salat (1980), Fridy (1985) and Connor (1988), rapid development were made on this subject. The notion of statistically convergent double sequences is studied by some authors (see for instance [5, 9, 10]).
Statistical convergence depends on the density of the subsets of N while the notion of statistically convergent double sequences depends on the density of subsets of N × N
Recall that a subset of N × N is said to have density () if the limit () = lim →∞ 1 X ≤ X ≤ ( )
exists. A double sequence () is said to be statistically convergent to in
Note that if () is a convergent double sequence then it is also statistically
convergent to the same number and if () is statistically convergent to a
number then this number is uniquely determined.
In this paper we introduce the notions of convergence and boundlessness of triple sequences. We also introduce and study statistical convergent triple sequences as well as the density of subsets of N × N × N.
2. Preliminaries and Definitions
Now we introduce some basic notions and examples related to the subject. Definition 1. A function : N × N × N → R (C) is called a real (complex) triple sequence, where N, R and C denote the sets of natural numbers, real numbers and complex numbers respectively.
Definition 2. A triple sequence () is said to be convergent to in
Pring-sheim’s sense if for every 0 there exists () ∈ N such that |− | whenever ≥ ≥ ≥ Example 1. Let = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ = 3 = 5 = 7 8 otherwise. Then () → 8 in Pringsheim’s sense.
Definition 3.A triple sequence () is said to be a Cauchy sequence if for
every 0 there exists () ∈ N such that
|− | whenever ≥ ≥ ≥ ≥ ≥ ≥
Definition 4. A triple sequence () is said to be bounded if there exists
0 such that || for all .
We denote the set of all bounded triple sequences by 3
∞ It can be easily shown
that 3
∞ is a normed space, normed by
kk(∞3)= sup
|| ∞
A subset of N × N is said to have natural density 2() if
2() = lim →∞
| ( )|
where the vertical bars denote the numbers of ( ) in such that ≤ and ≤ .
A subset of N × N × N is said to have natural density 3() if
3() = lim →∞
| ( )| exists,
where the vertical bars denote the numbers of ( ) in such that ≤ , ≤ and ≤ .
Example 2. Let =©¡3 3 3¢: ∈ Nª then 3() = lim →∞ |()| ≤ lim 3 √√3√3 = 0
Example 3. Let = {( 3 5) : ∈ N} then 3() = lim →∞
|()|
=
1 15.
Definition 5. A real triple sequence () is said to be statistically convergent
to the number if for each 0,
3({( ) ∈ N × N × N : |− | ≥ }) = 0
If () is statistically convergent to the number we denote this by
-lim
→∞=
Remark 1. It is clear that if () is convergent then it is -convergent but
the converse is not necessarily true.
It is clear from the following example that a statistically convergent triple se-quence may not be bounded.
Example 4. Let
=
½
are cubes 5 otherwise.
Then -lim = 5 but is neither convergent in Pringsheim’s sense nor
bounded.
3. Main Results
Theorem 1. If () is a real triple sequence then -lim = if and only if
there exists a subset ⊆ N× N×N such that 3() = 1 and lim →∞ ()∈
=
Proof. This theorem can be proved by using the similar way as in theorem 2 in [2].
Remark 2.If -lim = then there exists a triple sequence such that
lim
= and 3(( ) ∈ N × N × N : 6= ) = 0.
Theorem 2. The set ∩ 3
∞ is a closed linear subspace of the normed linear
space 3 ∞.
Proof. Let ()= ()
∈ ∩ 3∞and ()→ ∈ 3∞ Since ()∈ ∩ 3∞
then there exists real number such that
- lim () = ( = 1 2 )
On the other hand since ()→ ∈ 3
∞, there exists () ∈ N such that
¯ ¯ ¯(Ξ ) − () ¯ ¯ ¯ 3
for every Ξ ≥ ≥ , ≥ ≥ , ≥ ≥ , where || denotes the norm in a linear space. From the preceding theorem there exists 1 ⊆ N × N × N such
that 3(1) = 1 and
lim
→∞ ()∈1
() =
and there exists 2⊆ N × N × N such that 3(2) = 1 and
lim
→∞ ()∈2
(Ξ ) = Ξ
Since 3(1∩ 2) = 1 1∩2is not finite. Let us choose (1 2 3) ∈ 1∩2
then we have ¯ ¯ ¯(Ξ )123− Ξ ¯ ¯ ¯ 3 and ¯ ¯ ¯()123− ¯ ¯ ¯ 3
Hence, for each Ξ ≥ ≥ , ≥ ≥ , ≥ ≥ , we have |− Ξ| ≤ ¯ ¯ ¯()123− ¯ ¯ ¯ +¯¯¯()123− (Ξ ) 123 ¯ ¯ ¯ +¯¯¯(Ξ )123− Ξ ¯ ¯ ¯ 3+ 3+ 3
and this means is a Cauchy sequence and thus convergent, say lim
=
Now we want to show is statistically convergent to Because ()→ , for each 0 there exists 1() such that for ≥ 1()
¯ ¯ ¯() − ¯ ¯ ¯ 3
and because lim
= for each 0 there exists 2() such that for
≥ 2()
|− |
3 Finally since ()is statistically convergent to
there exists a subset =
{( )} ⊆ N × N × N such that 3() = 1 and for every 0 there is an
3() such that for all ≥ 3() and ( ) ∈
¯ ¯ ¯() − ¯ ¯ ¯ 3 If we let 4() = max {1() 2() 3()} then
|− | ≤ ¯ ¯ ¯() − ¯ ¯ ¯ +¯¯¯() − ¯ ¯ ¯ + |− |
So, is statistically convergent to and this completes the proof. Theorem 3. The set ∩ 3
∞ is nowhere dense in 3∞
Proof. We know that if is a linear normed space then its every closed linear subspace different from is nowhere dense in The following sequence, which is bounded but not -convergent, shows that ∩ 3
∞6= 3∞
=
½
−5 are odd 5 otherwise.
Now to show that every statistically convergent triple sequence is statistically Cauchy, we introduce the following definitions.
Definition 6. A triple sequence () is said to be statistically Cauchy
se-quence if for every 0 there exists = (), = (), = () ∈ N such that 3({( ) ∈ N × N × N : |− | ≥ }) = 0
Definition 7. Let = () and = () be two triple sequences. Then we
say that = for almost all if 3({( ) ∈ N × N × N : 6= }) =
0
Definition 8. Let = () be a triple sequence. A subset of C is said to
contain for almost all if 3({( ) ∈ N × N × N : ∈ }) = 0
where C denotes the set of complex numbers.
Theorem 4. A triple sequence () is statistically convergent if and only if
it is statistically Cauchy.
Proof: Necessity is obvious. To prove sufficiency let () be a statistically
Cauchy sequence. Let = 1 then there exists 1 1 1 such that the closed
Now for = 12 there exists 2 2 2 such that the closed ball 2 of
diameter 1 unit with centre at 222 contains for almost all If
we take 2 = 1∩ 2 then 2 is a closed subset of C with diameter less than
or equal to 1 unit such that 2 contains for almost all If we take
= 1
22 then exists 3 3 3 such that the closed ball 3 of diameter 12 unit
with centre at 333contains for almost all Say 3= 2∩
3then
3 is a closed subset of C with diameter less than or equal to 12 unit such that
3 contains for almost all Following this way, we have a sequence
() of closed subsets of C such that
(i) +1⊆ for all ∈ N
(ii) ≤ 22− for all ∈ N
Then
∞
\
=1
contains one point. Denote this point by . Then ∈ for all
∈ N If we choose such that 21 then contains for almost all
This means is statistically convergent to
Now the following theorem is easy to prove.
Theorem. 5.:Let is a triple sequence then the following statements are
equivalent:
(i) is statistically convergent to ;
(ii) is statistically Cauchy;
(iii) there exists a subsequence of such that lim
=
References
1.H. Fast, Sur la convergence statistique, Colloq. Math.,2 (1951) 241 − 244 2.J.A. Fridy, Statistical limit points, Proc. Amer. Math. Soc., 118 (1993) 1187 − 1192.
3.J.A. Fridy, and C. Orhan, Statistical limit superior and inferior, Proc. Amer. Math. Soc.,125 (1997) 3625 − 3631.
4.H.J. Hamilton, Transformations of multiple sequences, Duke Math. J., 2 (1936) 29 − 60
5.O. Mursaleen, and H.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl.,288 (2003) 223 − 231
6.A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53 (1900) 289 − 321
7.G.M. Robinson, Divergent double sequences and series, Trans. Amer. Math. Soc., 28 (1926) 50 − 73
8.H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math.,2 (1951) 73 − 74
9.B.C. Tripathy, Statistically convergent double sequences, Tamkang J. Math.,34 (3) (2003) 231 − 237
10.B.C. Tripathy, On-convergent double sequences, Soochow J. of Math.,31 (4) (2005) 549 − 560.
11.I.J. Schoenberg, The intregrability of certain functions and related summability methods, Amer. Math. Monthly,66 (1959) 361 − 375
