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Weighted -Statistical Convergence for Sequences of Positive Linear Operators
We introduce the notion of weighted -statistical convergence of a sequence, where represents the nonnegative regular matrix. We also prove the Korovkin approximation theorem by using the notion of weighted -statistical convergence. Further, we give a rate of weighted -statistical convergence and apply the classical Bernstein polynomial to construct an illustrative example in support of our result.
1. Background, Notations, and Preliminaries
We begin this paper by recalling the definition of natural (or asymptotic) density as follows. Suppose that and . Then is called the natural density of provided that the limit exists, where represents the number of elements in the enclosed set.
The term “statistical convergence” was first presented by Fast  which is a generalization of the concept of ordinary convergence. Actually, a root of the notion of statistical convergence can be detected by Zygmund  (also, see ), where he used the term “almost convergence” which turned out to be equivalent to the concept of statistical convergence. The notion of Fast was further investigated by Schoenberg , Šalát , Fridy , and Conner .
The following notion is due to Fast . A sequence is said to be statistically convergent to if for every , where Equivalently, In symbol, we will write . We remark that every convergent sequence is statistically convergent but not conversely.
Let and be two sequence spaces and let be an infinite matrix. If for each in the series converges for each and the sequence belongs to , then we say that matrix maps into . By the symbol we denote the set of all matrices which map into .
A matrix (or a matrix map ) is called regular if , where the symbol denotes the spaces of all convergent sequences and for all . The well-known Silverman-Toeplitz theorem (see ) asserts that is regular if and only if(i) for each ;(ii);(iii).
Kolk  extended the definition of statistical convergence with the help of nonnegative regular matrix calling it -statistical convergence. The definition of -statistical convergence is given by Kolk as follows. For any nonnegative regular matrix , we say that a sequence is -statistically convergent to provided that for every we have
In 2009, the concept of weighted statistical convergence was defined and studied by Karakaya and Chishti  and further modified by Mursaleen et al.  in 2012. In 2013, Belen and Mohiuddine  presented a generalization of this notion through de la Vallée-Poussin mean. Quite recently, Esi  defined and studied the notion statistical summability through de la Vallée-Poussin mean in probabilistic normed spaces.
Let be a sequence of nonnegative numbers such that and Let We say that the sequence is -summable to if .
The lower and upper weighted densities of are defined by respectively. We say that has weighted density, denoted by , if the limits of both of the above densities exist and are equal; that is, one writes
The sequence is said to be weighted statistically convergent (or ) to if, for every , the set has weighted density zero; that is, In this case we write .
Remark 1. If for all , then -summable is reduced to -summable (or Cesàro summable) and weighted statistical convergence is reduced to statistical convergence.
On the other hand, let us recall that is the space of all functions continuous on . We know that is a Banach space with norm Suppose that is a linear operator from into . It is clear that if implies , then the linear operator is positive on . We denote the value of at a point by . The classical Korovkin approximation theorem states the following .
Theorem 2. Let be a sequence of positive linear operators from into . Then, for all if and only if where and .
Many mathematicians extended the Korovkin-type approximation theorems by using various test functions in several setups, including Banach spaces, abstract Banach lattices, function spaces, and Banach algebras. Firstly, Gadjiev and Orhan  established classical Korovkin theorem through statistical convergence and display an interesting example in support of our result. Recently, Korovkin-type theorems have been obtained by Mohiuddine  for almost convergence. Korovkin-type theorems were also obtained in  for -statistical convergence. The authors of  established these types of approximation theorem in weighted spaces, where , through -summability which is stronger than ordinary convergence. For these types of approximation theorems and related concepts, one can be referred to [19–27] and references therein.
2. Korovkin-Type Theorems by Weighted -Statistical Convergence
Kolk  introduced the notion of -statistical convergence by considering nonnegative regular matrix instead of Cesáro matrix in the definition of statistical convergence due to Fast. Inspired from the work of Kolk, we introduce the notion of weighted -statistical convergence of a sequence and then we establish some Korovkin-type theorems by using this notion.
Definition 3. Let be a nonnegative regular matrix. A sequence of real or complex numbers is said to be weighted -statistically convergent, denoted by -convergent, to if for every where In symbol, we will write .
Remark 4. One has the following.(i)If we take , where denotes the identity matrix, then weighted -statistical convergence of a sequence is reduced to ordinary convergence.(ii)If we take , where denotes the Cesáro matrix of order one, then weighted -statistical convergence of a sequence reduces to weighted statistical convergence.(iii)If we take and for all , then weighted -statistical convergence of a sequence reduces to statistical convergence.
Note that convergent sequence implies weighted -statistically convergent to the same value but the converse is not true in general. For example, take and for all and define a sequence by where . Then this sequence is statistically convergent to 0 but not convergent; in this case, weighted -statistical convergence of a sequence coincides with statistical convergence.
Theorem 5. Let be a nonnegative regular matrix. Consider a sequence of positive linear operators from into itself. Then, for all bounded on whole real line, if and only if
Proof. Equation (20) directly follows from (19) because each of belongs to . Consider a function . Then there is a constant such that for all . Therefore, Let be given. By hypothesis there is a such that Solving (21) and (22) and then substituting , one obtains Equation (23) can also be written as Operating to (24) since is linear and monotone, one obtains Note that is fixed, so is constant number. Thus, we obtain from (25) that The term “” in (26) can also be written as Now substituting the value of in (26), we get that We can rewrite the term “” in (28) as follows: Equation (28) with the above value of becomes Therefore, where . Taking supremum over , one obtains or where Hence, For a given , choose such that , and we will define the following sets: It is easy to see that Thus, for each , we obtain from (35) that Taking limit in (38) and also (20) gives that This yields that for all .
We also obtain the following Korovkin-type theorem for weighted statistical convergence by writing Cesáro matrix instead of nonnegative regular matrix in Theorem 5.
Theorem 6. Consider a sequence of positive linear operators from into itself. Then, for all if and only if
Remark 8. By Theorem 2 of , we have that if a sequence is weighted statistically convergent to , then it is strongly -summable to provided that is bounded; that is, there exists a constant such that for all . We write for the set of all sequences which are strongly -summable to .
Proof. It follows from (49) that , for some constant and for all . Hence, for , one obtains Right hand side of (51) is constant, so is bounded. Since (49) implies (42), by Theorem 6 we get that By Remark 8, (51) and (52) together give the desired result.
3. Rate of Weighted -Statistical Convergence
First we define the rate of weighted -statistically convergent sequence as follows.
Definition 10. Let be a nonnegative regular matrix and let be a positive nonincreasing sequence. Then, a sequence is weighted -statistically convergent to with the rate of if for each where In symbol, we will write
We will prove the following auxiliary result by using the above definition.
Lemma 11. Let be a nonnegative regular matrix. Suppose that and are two positive nonincreasing sequences. Let and be two sequences such that and . Then,(i),(ii),(iii), for any scalar ,where .
Proof. (i) Suppose that Given , define It is easy to see that This yields that holds for all . Since , (59) gives that Taking limit in (60) together with (56), we obtain Thus, Similarly, we can prove (ii) and (iii).
Recall that the modulus of continuity of in is defined by It is well known that
Theorem 12. Let be a nonnegative regular matrix. If the sequence of positive linear operators satisfies the conditions(i),(ii) with and ,where and are two positive nonincreasing sequences, then for all , where .
Proof. Equation (27) can be reformed into the following form: Choosing , one obtains where . For a given , we will define the following sets: It follows from (67) that holds for . Since , we obtain from (69) that Taking limit in (70) together with Lemma 11 and our hypotheses (i) and (ii), one obtains This yields
4. Example and the Concluding Remark
The operators given by where are the fundamental Bernstein polynomials defined by for any , any , and any , are called Bernstein operators and were first introduced in . Let the sequence be defined by with , where is a sequence defined by That is, . Let and consider a nonnegative regular matrix . Then, Since is weighted statistically convergent to 0 but not convergent. It is not difficult to see that and the sequence satisfies conditions (20). This yields that
On the other hand, one obtains , since , and hence It follows that does not satisfy the Korovkin theorem, since and hence is not convergent. Finally, we conclude that Theorem 5 is stronger than Theorem 2.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (351/130/1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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