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On pointwise and uniform statistical convergence of order α for sequences of functions

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 S α (f)-statistical convergence and strong w p β (f)-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. [79], Fridy [10], Güngör et al. [11], Işık [12, 13], Kolk [14], Mohiuddine et al. [1519], 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 w p β (f) and S α (f) and between S α (f) and S(f).

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 of natural numbers. The density of a subset E of is defined by

where χ E is the characteristic function of E. It is clear that any finite subset of has zero natural density and δ( E c )=1δ(E).

The α-density of a subset E of was defined by Çolak [28]. Let α be a real number such that 0<α1. The α-density of a subset E of is defined by

δ α (E)= lim n 1 n α |{kn:kE}|provided the limit exists,

where |{kn:kE}| denotes the number of elements of E not exceeding n.

If x=( x k ) is a sequence such that x k satisfies property P(k) for almost all k except a set of α-density zero, then we say that x k satisfies property P(k) for ‘almost all k according to α’ and we abbreviate this by ‘a.a.k (α)’.

It is clear that any finite subset of has zero α density and δ α ( E c )=1 δ α (E) does not hold for 0<α<1 in general, the equality holds only if α=1. Note that the α-density of any set reduces to the natural density of the set in case α=1.

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 0<α1 be given. The sequence ( x k ) is said to be statistically convergent of order α if there is a real number such that

lim n 1 n α | { k n : | x k | ε } |=0,

for every ε>0, in which case we say that x is statistically convergent of order α to . In this case, we write S α lim x k =. The set of all statistically convergent sequences of order α will be denoted by S α . We write S 0 α to denote the set of all statistically null sequences of order α. It is clear that S 0 α S α for each 0<α1. The statistical convergence of order α is same with the statistical convergence for α=1.

A sequence x=( x k ) is said to be strongly Cesàro summable to a number if lim n 1 n × k = 1 n | x k |=0. The set of strongly Cesàro summable sequences is denoted by [C,1] and defined as

[C,1]= { x = ( x k ) : lim n 1 n k = 1 n | x k | = 0  for some  } .

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 αβ.

Definition 3.1 Let 0<α1 be given. A sequence of functions { f k } 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 ε>0,

lim n 1 n α | { k n : | f k ( x ) f ( x ) | ε  for every  x A } |=0

i.e., for every xA,

| f k (x)f(x)|<ε a.a.k (α).
(1)

In this case, we write S α lim f k (x)=f(x) on A. S α lim f k (x)=f(x) means that for every δ>0 and 0<α1, there is an integer N such that

1 n α | { k n : | f k ( x ) f ( x ) | ε  for every  x A } |<δ

for all n>N (=N(ε,δ,x)) and for each ε>0. The set of all pointwise statistically convergent sequences of functions order α will be denoted by S α (f). For α=1, we will write S(f) instead of S α (f) and in the special case f=0, we will write S 0 α (f) instead of S α (f).

The statistical convergence of order α for a sequence of functions is well defined for 0<α1. But it is not well defined for α>1. For this, let { f k } be defined as follows:

f k (x)={ 1 , k = 2 n , x k , k 2 n , n=1,2,3,,x [ 0 , 1 2 ] .

Then both

lim n 1 n α | { k n : | f k ( x ) 1 | ε  for every  x A } |= lim n n 2 n α =0

and

lim n 1 n α | { k n : | f k ( x ) 0 | ε  for every  x A } |= lim n n 2 n α =0

for α>1, so that { f k } statistically converges of order α both to 1 and 0, i.e., S α lim f k (x)=1 and S α lim f k (x)=0, which is impossible.

Theorem 3.2 Let 0<α1 and { f k }, { g k } be sequences of real-valued functions defined in a set A.

  1. (i)

    If S α lim f k (x)=f(x) and cR, then S α limc f k (x)=cf(x).

  2. (ii)

    If S α lim f k (x)=f(x) and S α lim g k (x)=g(x), then S α lim( f k (x)+ g k (x))=f(x)+g(x).

Proof (i) The proof is clear in case c=0. Suppose that c0 and S α lim f k (x)=f(x), then there exists ε>0 such that

| f k (x)f(x)|< ε | c | a.a.k (α),

and hence

|c f k (x)cf(x)|<ε a.a.k (α).

This implies that S α limc f k (x)=cf(x).

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, c(f) S α (f) for each 0<α1. But the converse of this does not hold. For example, the sequence { f k } defined by

f k (x)={ 1 , k = n 3 , 2 k x 1 + k 2 x 2 , k n 3

is statistically convergent of order α with S α lim f k (x)=0 for α> 1 3 , but it is not convergent. □

Definition 3.3 Let α be any real number such that 0<α1 and let { f k } be a sequence of functions on a set A. The sequence { f k } is a statistically Cauchy sequence of order α (or α-statistically Cauchy sequence) provided that for every ε>0, there exists a number N (=N(ε,x)) such that

| f k (x) f N (x)|<ε a.a.k (α),

i.e.,

lim n 1 n α | { k n : | f k ( x ) f N ( x ) | ε  for every  x A } |=0.

Theorem 3.4 Let { f k } be a sequence of functions defined on a set A. The following statements are equivalent:

  1. (i)

    { f k } is a pointwise α-statistically convergent sequence on A;

  2. (ii)

    { f k } is a α-statistically Cauchy sequence on A;

  3. (iii)

    { f k } is a sequence of functions for which there is a pointwise convergent sequence of order α, a sequence of functions { g k } such that f k (x)= g k (x) a.a.k (α) for every xA.

Proof (i) (ii) Suppose that S α lim f k (x)=f(x) on A and let ε>0. Then | f k (x)f(x)|< ε 2 a.a.k (α) and if N is chosen so that | f N (x)f(x)|< ε 2 , then we have

| f k (x) f N (x)|| f k (x)f(x)|+| f N (x)f(x)|< ε 2 + ε 2 a.a.k (α)

for every xA. Hence { f k } is an α-statistically Cauchy sequence.

Next, assume (ii) is true and choose N so that the band I=[ f N (x)1, f N (x)+1] contains f k (x) a.a.k (α) for every xA. Also, apply (ii) to choose M so that I =[ f M (x) 1 2 , f M (x)+ 1 2 ] contains f k (x) a.a.k (α) for every xA. We assert that

I 1 =I I  contains  f k (x) a.a.k (α)for every xA;

for

so

Therefore, I 1 is a closed band of height less than or equal to 1 that contains f k (x) a.a.k (α) for every xA. Now we proceed by choosing N(2) so that I =[ f N ( 2 ) (x) 1 4 , f N ( 2 ) (x)+ 1 4 ] contains f k (x) a.a.k (α), and by the preceding argument, I 2 = I 1 I contains f k (x) a.a.k (α) for every xA and I 2 has height less than or equal to 1 2 . Continuing inductively, we construct a sequence { I m } m = 1 of closed band such that for each m, I m I m + 1 , the height of I m is not greater than 2 1 m and f k (x) I m a.a.k (α) for every xA. Thus there exists a function f(x), defined on A, such that {f(x)} is equal to m = 1 I m . Using the fact that f k (x) I m a.a.k (α) for every xA, we choose an increasing positive integer sequence { T m } m = 1 such that

1 n α | { k n : f k ( x ) I m  for every  x A } |< 1 m if n> T m .
(2)

Now define a subsequence ( z k (x)) of ( f k (x)) consisting of all terms f k (x) such that k> T 1 and if T m <k T m + 1 then f k (x) I m for every xA. Next, define the sequence of functions ( g k (x)) by

g k (x)={ f ( x ) if  f k ( x )  is a term of  z k ( x ) , f k ( x )  otherwise

for every xA. Then lim k g k (x)=f(x) on A; for if ε> 1 m >0 and k> T m , then either f k (x) is a term of ( z k (x)) or g k (x)= f k (x) I m on A and | g k (x) f k (x)| height of I m 2 1 m for every xA. We also assert that g k (x)= f k (x) a.a.k (α) for every xA. To verify this, we observe that if T m <n T m + 1 , then

So, by (2)

Hence, the limit is 0 as n and f k (x)= g k (x) a.a.k (α) for every xA. Therefore, (ii) implies (iii).

Finally, assume that (iii) holds, say f k (x)= g k (x) a.a.k (α) for every xA and lim k g k (x)=f(x) on A. Let ε>0. Then for each n,

since lim k g k (x)=f(x) on A, the latter set contains a fixed number of integers, say l=l(ε,x). Therefore,

because f k (x)= g k (x) a.a.k (α) for every xA. Hence | f k (x)f(x)|<ε a.a.k (α) for every xA, so (i) holds and the proof is complete. □

Corollary 3.5 If { f k } is a sequence of functions such that S α lim f k (x)=f(x) on A, then { f k } has a subsequence { f k ( n ) (x)} such that lim n f k ( n ) (x)=f(x) on A.

Theorem 3.6 Let 0<αβ1. Then S α (f) S β (f) and the inclusion is strict for some α and β such that α<β.

Proof If 0<αβ1, then

for every ε>0 and this gives that S α (f) S β (f). To show that the inclusion is strict, consider the sequence { f k } defined by

f k (x)={ 1 , k = n 2 , k 2 x 1 + k 3 x 2 , k n 2 , n=1,2,3,,x[0,1].

Hence we can write for 1 2 <α1

Then S β lim f k (x)=0, i.e., x S β (f) for 1 2 <β1, but x S α (f) for 0<α 1 2 . □

If we take β=1 in Theorem 3.6, then we obtain the following result.

Corollary 3.7 If a sequence of functions { f k } is statistically convergent of order α, to the function f for some 0<α1, then it is statistically convergent to the function f.

Definition 3.8 Let α be any real number such that 0<α1 and let p be a positive real number. A sequence of functions { f k } is said to be strongly p-Cesàro summable of order α if there is a function f such that

lim n 1 n α k = 1 n | f k (x)f(x) | p =0.

In this case, we write w p α lim f k (x)=f(x) on A. The strong p-Cesàro summability of order α reduces to the strong p-Cesàro summability for α=1. The set of all strongly p-Cesàro summable sequences of functions of order α will be denoted by w p α (f). We write w o , p α (f) in case f(x)=0.

Theorem 3.9 Let 0<αβ1 and p be a positive real number. Then w p α (f) w p β (f) and the inclusion is strict for some α and β such that α<β.

Proof Let the sequence { f k } be strongly p-Cesàro summable of order α. Then, given α and β such that 0<αβ1 and a positive real number p, we may write

1 n β k = 1 n | f k (x)f(x) | p 1 n α k = 1 n | f k (x)f(x) | p ,

and this gives that w p α (f) w p β (f).

To show that the inclusion is strict, consider the sequence { f k } defined by

f k (x)={ 1 1 + k x , k = n 2 , 0 , k n 2 , x [ 0 , 1 k ] .

Then

1 n β k = 1 n | f k (x)0 | p n n β = 1 n β 1 2

since 1/( n β 1 2 )0 as n, then w p β lim f k (x)=0, i.e., the sequence { f k } is strongly p-Cesàro summable of order α for 1 2 <β1, but since

n 2 n α 1 n α k = 1 n | f k (x)0 | p

and n /2 n α , n, the sequence { f k } is not strongly p-Cesàro summable of order α for 0<α< 1 2 . □

Corollary 3.10 Let 0<αβ1 and p be a positive real number. Then

  1. (i)

    if α=β, then w p α (f)= w p β (f);

  2. (ii)

    w p α (f) w p (f) for each α(0,1] and 0<p<.

Theorem 3.11 Let 0<α1 and 0<p<q<. Then w q α (f) w p α (f).

Proof Omitted. □

Theorem 3.12 Let α and β be fixed real numbers such that 0<αβ1 and 0<p<. If a sequence of functions { f k } 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 { f k } defined on A, we can write

k = 1 n | f k (x)f(x) | p | { k n : | f k ( x ) f ( x ) | ε  for every  x A } | ε p

and so that

1 n α k = 1 n | f k ( x ) f ( x ) | p 1 n α | { k n : | f k ( x ) f ( x ) | ε  for every  x A } | ε p 1 n β | { k n : | f k ( x ) f ( x ) | ε  for every  x A } | ε p .

 □

Corollary 3.13 Let α be a fixed real number such that 0<α1 and 0<p<. If a sequence of functions { f k } 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 0<α1. A sequence of functions { f k } 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 ε>0,

lim n 1 n α | { k n : | f k ( x ) f ( x ) | ε  for all  x A } |=0,

i.e., for all xA,

| f k (x)f(x)|<ε a.a.k (α).
(3)

In this case, we write

S α lim f k (x)=f(x)uniformly on A or  S u α lim f k (x)=f(x) on A.

The set of all uniformly α-statistically convergent sequences will be denoted by S u α (f).

Theorem 3.15 Let f and f k , for all kN, be continuous functions on A=[a,b]R and 0<α1. Then S α lim f k (x)=f(x) uniformly on A if and only if S α lim c k =0, where c k = max x A | f k (x)f(x)|.

Proof Suppose that S α lim f k (x)=f(x) uniformly on A. Since | f k (x)f(x)| is continuous on A for each kN, it has absolute maximum value at some point x k A, i.e., there exist x 1 , x 2 ,A such that c 1 =| f 1 ( x 1 )f( x 1 )|, c 2 =| f 2 ( x 2 )f( x 2 )|, , etc. Thus we may write c k =| f k ( x k )f( x k )|, k=1,2, . From the definition of uniform α-statistical convergence, we may write, for every ε>0,

| f k ( x k )f( x k )|<ε a.a.k (α).

Hence, S α lim c k =0.

The necessity is trivial. □

It follows from (3) that if lim f k (x)=f(x) uniformly on A, then S α lim f k (x)=f(x) uniformly on A. But the converse is not true, for this consider the sequence defined by

f k (x)={ 1 , k = n 2 , k k 2 + k 2 x 2 otherwise , k=1,2,3,,x[0,1].

Then if x[0,1] and α[ 1 2 ,1], then { f k } is uniformly α-statistically convergent to f(x)=0 on [0,1] since S α lim c k =0, where

c k = max x [ 0 , 1 ] | f k (x)0|={ 2 , k = n 2 , 1 k otherwise ,

but ( f k (x)) is not uniformly convergent on [0,1] since lim k c k does not exist.

Corollary 3.16

  1. (i)

    lim f k (x)=f(x) uniformly on Alim f k (x)=f(x) on A S α lim f k (x)=f(x) pointwise on A.

  2. (ii)

    S α lim f k (x)=f(x) uniformly on A S α lim f k (x)=f(x) pointwise on A.

  3. (iii)

    If 0<αβ1, then S u α (f) S u β (f).

Definition 3.17 Let α be any real number such that 0<α1 and let { f k } be a sequence of functions on a set A. The sequence { f k } is a uniformly statistically Cauchy sequence of order α (or uniformly α-statistically Cauchy sequence) provided that for every ε>0, there exists a number N (=N(ε)) such that

| f k (x) f N (x)|<ε a.a.k (α) for all xA,

i.e.,

lim n 1 n α | { k n : | f k ( x ) f N ( x ) | ε  for all  x A } |=0.

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 0<α1 and { f k }, { g k } be sequences of real-valued functions defined on a set A.

  1. (i)

    If S u α lim f k (x)=f(x) and cR, then S u α limc f k (x)=cf(x).

  2. (ii)

    If S u α lim f k (x)=f(x) and S u α lim g k (x)=g(x), then S u α lim( f k (x)+ g k (x))=f(x)+g(x).

Theorem 3.19 Let α be any real number such that 0<α1 and let { f k } be a sequence of functions on a set A. The following statements are equivalent:

  1. (i)

    { f k } is a uniformly α-statistically convergent sequence on A;

  2. (ii)

    { f k } is a uniformly α-statistically Cauchy sequence on A;

  3. (iii)

    { f k } is a sequence of functions for which there is a uniformly convergent sequence of order α, a sequence of functions { g k } such that f k (x)= g k (x) a.a.k (α) for all xA.

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Correspondence to Muhammed Çinar.

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The authors declare that they have no competing interests.

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MC, MK, and ME have contributed to all parts of the article. All authors read and approved the final manuscript.

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Keywords

  • statistical convergence
  • sequences of functions
  • Cesàro summability