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On pointwise and uniform statistical convergence of order α for sequences of functions
Fixed Point Theory and Applications volume 2013, Article number: 33 (2013)
Abstract
In this paper, we introduce the concepts of pointwise and uniform statistical convergence of order α for sequences of real-valued functions. Furthermore, we give the concept of an α-statistically Cauchy sequence for sequences of real-valued functions and prove that it is equivalent to pointwise statistical convergence of order α for sequences of real-valued functions. Also, some relations between -statistical convergence and strong -summability are given.
MSC:40A05, 40C05, 46A45.
1 Introduction
The idea of statistical convergence was given by Zygmund [1] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [2] and Fast [3] and later reintroduced by Schoenberg [4] independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Başar [5], Connor [6], Et et al. [7–9], Fridy [10], Güngör et al. [11], Işık [12, 13], Kolk [14], Mohiuddine et al. [15–19], Miller and Orhan [20], Mursaleen [21], Rath and Tripathy [22], Salat [23], Savaş [24] 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 generalizations are also connected with subsets of the Stone-Čech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability.
The definitions of pointwise and uniform statistical convergence of sequences of real-valued functions were given by Gökhan et al. [25, 26] and independently by Duman and Orhan [27]. In the present paper, we introduce and examine the concepts of pointwise and uniform statistical convergence of order α for sequences of real-valued functions. In Section 2 we give a brief overview of statistical convergence of order α and strong p-Cesàro summability. In Section 3 we give the concepts of pointwise and uniform statistical convergence of order α, and the concept α-statistically Cauchy sequence for sequences of real-valued functions and prove that it is equivalent to pointwise statistical convergence of order α for sequences of real-valued functions. We also establish some inclusion relations between and and between and .
2 Definition and preliminaries
The definitions of statistical convergence and strong p-Cesàro convergence of a sequence of real numbers were introduced in the literature independently of one another and have followed different lines of development since their first appearance. It turns out, however, that the two definitions can be simply related to one another in general and are equivalent for bounded sequences. The idea of statistical convergence depends on the density of subsets of the set ℕ of natural numbers. The density of a subset E of ℕ is defined by
where is the characteristic function of E. It is clear that any finite subset of ℕ has zero natural density and .
The α-density of a subset E of ℕ was defined by Çolak [28]. Let α be a real number such that . The α-density of a subset E of ℕ is defined by
where denotes the number of elements of E not exceeding n.
If is a sequence such that satisfies property for almost all k except a set of α-density zero, then we say that satisfies property for ‘almost all k according to α’ and we abbreviate this by ‘a.a.k ’.
It is clear that any finite subset of ℕ has zero α density and does not hold for in general, the equality holds only if . Note that the α-density of any set reduces to the natural density of the set in case .
The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan in [29], and after then statistical convergence of order α and strong p-Cesàro summability of order α were studied by Çolak [28].
The statistical convergence of order α is defined as follows. Let be given. The sequence is said to be statistically convergent of order α if there is a real number ℓ such that
for every , in which case we say that x is statistically convergent of order α to ℓ. In this case, we write . The set of all statistically convergent sequences of order α will be denoted by . We write to denote the set of all statistically null sequences of order α. It is clear that for each . The statistical convergence of order α is same with the statistical convergence for .
A sequence is said to be strongly Cesàro summable to a number ℓ if . The set of strongly Cesàro summable sequences is denoted by and defined as
There is a natural relationship between statistical convergence and strong p-Cesàro summability.
3 Main result
In this section we give the main results of this article. We give relations between the statistical convergence of order α and the statistical convergence of order β for sequences of functions, the relations between the strong p-Cesàro summability of order α and the strong p-Cesàro summability of order β and the relations between the strong p-Cesàro summability of order α and the statistical convergence of order β for sequences of real-valued functions, where .
Definition 3.1 Let be given. A sequence of functions is said to be pointwise statistically convergent of order α (or pointwise α-statistically convergent sequence) to the function f on a set A if, for every ,
i.e., for every ,
In this case, we write on A. means that for every and , there is an integer N such that
for all () and for each . The set of all pointwise statistically convergent sequences of functions order α will be denoted by . For , we will write instead of and in the special case , we will write instead of .
The statistical convergence of order α for a sequence of functions is well defined for . But it is not well defined for . For this, let be defined as follows:
Then both
and
for , so that statistically converges of order α both to 1 and 0, i.e., and , which is impossible.
Theorem 3.2 Let and , be sequences of real-valued functions defined in a set A.
-
(i)
If and , then .
-
(ii)
If and , then .
Proof (i) The proof is clear in case . Suppose that and , then there exists such that
and hence
This implies that .
The proof of (ii) follows from the following inequalities:
It is easy to see that every convergent sequence of functions is statistically convergent of order α, that is, for each . But the converse of this does not hold. For example, the sequence defined by
is statistically convergent of order α with for , but it is not convergent. □
Definition 3.3 Let α be any real number such that and let be a sequence of functions on a set A. The sequence is a statistically Cauchy sequence of order α (or α-statistically Cauchy sequence) provided that for every , there exists a number N () such that
i.e.,
Theorem 3.4 Let be a sequence of functions defined on a set A. The following statements are equivalent:
-
(i)
is a pointwise α-statistically convergent sequence on A;
-
(ii)
is a α-statistically Cauchy sequence on A;
-
(iii)
is a sequence of functions for which there is a pointwise convergent sequence of order α, a sequence of functions such that a.a.k for every .
Proof (i) ⇒ (ii) Suppose that on A and let . Then a.a.k and if N is chosen so that , then we have
for every . Hence is an α-statistically Cauchy sequence.
Next, assume (ii) is true and choose N so that the band contains a.a.k for every . Also, apply (ii) to choose M so that contains a.a.k for every . We assert that
for
so
Therefore, is a closed band of height less than or equal to 1 that contains a.a.k for every . Now we proceed by choosing so that contains a.a.k , and by the preceding argument, contains a.a.k for every and has height less than or equal to . Continuing inductively, we construct a sequence of closed band such that for each m, , the height of is not greater than and a.a.k for every . Thus there exists a function , defined on A, such that is equal to . Using the fact that a.a.k for every , we choose an increasing positive integer sequence such that
Now define a subsequence of consisting of all terms such that and if then for every . Next, define the sequence of functions by
for every . Then on A; for if and , then either is a term of or on A and height of for every . We also assert that a.a.k for every . To verify this, we observe that if , then
So, by (2)
Hence, the limit is 0 as and a.a.k for every . Therefore, (ii) implies (iii).
Finally, assume that (iii) holds, say a.a.k for every and on A. Let . Then for each n,
since on A, the latter set contains a fixed number of integers, say . Therefore,
because a.a.k for every . Hence a.a.k for every , so (i) holds and the proof is complete. □
Corollary 3.5 If is a sequence of functions such that on A, then has a subsequence such that on A.
Theorem 3.6 Let . Then and the inclusion is strict for some α and β such that .
Proof If , then
for every and this gives that . To show that the inclusion is strict, consider the sequence defined by
Hence we can write for
Then , i.e., for , but for . □
If we take in Theorem 3.6, then we obtain the following result.
Corollary 3.7 If a sequence of functions is statistically convergent of order α, to the function f for some , then it is statistically convergent to the function f.
Definition 3.8 Let α be any real number such that and let p be a positive real number. A sequence of functions is said to be strongly p-Cesàro summable of order α if there is a function f such that
In this case, we write on A. The strong p-Cesàro summability of order α reduces to the strong p-Cesàro summability for . The set of all strongly p-Cesàro summable sequences of functions of order α will be denoted by . We write in case .
Theorem 3.9 Let and p be a positive real number. Then and the inclusion is strict for some α and β such that .
Proof Let the sequence be strongly p-Cesàro summable of order α. Then, given α and β such that and a positive real number p, we may write
and this gives that .
To show that the inclusion is strict, consider the sequence defined by
Then
since as , then , i.e., the sequence is strongly p-Cesàro summable of order α for , but since
and , , the sequence is not strongly p-Cesàro summable of order α for . □
Corollary 3.10 Let and p be a positive real number. Then
-
(i)
if , then ;
-
(ii)
for each and .
Theorem 3.11 Let and . Then .
Proof Omitted. □
Theorem 3.12 Let α and β be fixed real numbers such that and . If a sequence of functions is strongly p-Cesàro summable of order α to the function f, then it is statistically convergent of order β to the function f.
Proof For any sequence of functions defined on A, we can write
and so that
□
Corollary 3.13 Let α be a fixed real number such that and . If a sequence of functions is strongly p-Cesàro summable of order α to the function f, then it is statistically convergent of order α to the function f.
Definition 3.14 Let α be any real number such that . A sequence of functions is said to be uniformly statistically convergent of order α or uniformly ( α-statistically convergent sequence) to the function f on a set A if, for every ,
i.e., for all ,
In this case, we write
The set of all uniformly α-statistically convergent sequences will be denoted by .
Theorem 3.15 Let f and , for all , be continuous functions on and . Then uniformly on A if and only if , where .
Proof Suppose that uniformly on A. Since is continuous on A for each , it has absolute maximum value at some point , i.e., there exist such that , , etc. Thus we may write , . From the definition of uniform α-statistical convergence, we may write, for every ,
Hence, .
The necessity is trivial. □
It follows from (3) that if uniformly on A, then uniformly on A. But the converse is not true, for this consider the sequence defined by
Then if and , then is uniformly α-statistically convergent to on since , where
but is not uniformly convergent on since does not exist.
Corollary 3.16
-
(i)
uniformly on on pointwise on A.
-
(ii)
uniformly on pointwise on A.
-
(iii)
If , then .
Definition 3.17 Let α be any real number such that and let be a sequence of functions on a set A. The sequence is a uniformly statistically Cauchy sequence of order α (or uniformly α-statistically Cauchy sequence) provided that for every , there exists a number N () such that
i.e.,
The proofs of the following two theorems are similar to those of Theorem 3.2 and Theorem 3.4, therefore we give them without proof.
Theorem 3.18 Let and , be sequences of real-valued functions defined on a set A.
-
(i)
If and , then .
-
(ii)
If and , then .
Theorem 3.19 Let α be any real number such that and let be a sequence of functions on a set A. The following statements are equivalent:
-
(i)
is a uniformly α-statistically convergent sequence on A;
-
(ii)
is a uniformly α-statistically Cauchy sequence on A;
-
(iii)
is a sequence of functions for which there is a uniformly convergent sequence of order α, a sequence of functions such that a.a.k for all .
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Çinar, M., Karakaş, M. & Et, M. On pointwise and uniform statistical convergence of order α for sequences of functions. Fixed Point Theory Appl 2013, 33 (2013). https://doi.org/10.1186/1687-1812-2013-33
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DOI: https://doi.org/10.1186/1687-1812-2013-33