#### Abstract

Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, , and in the space as well as for the functions 1, cos, and sin in the space of all continuous 2-periodic functions on the real line. In this paper, we use the notion of -statistical -summability to prove the Korovkin second approximation theorem. We also study the rate of -statistical -summability of a sequence of positive linear operators defined from into .

#### 1. Introduction and Preliminaries

Let be the set of all natural numbers, , and . Then the natural density of is defined by if the limit exists, where the vertical bars indicate the number of elements in the enclosed set, is the Cesàro matrix of order , and denotes the characteristic sequence of given by

A sequence is said to be statistically convergent to if for every , the set has natural density zero (cf. Fast [1]); that is, for each , In this case, we write By the symbol we denote the set of all statistically convergent sequences. Statistical convergence of double sequences is studied in [2, 3].

A matrix is called regular if it transforms a convergent sequence into a convergent sequence leaving the limit invariant. The well-known necessary and sufficient conditions (Silverman-Toeplitz) for to be regular are(i); (ii), for each ;(iii).

Freedmann and Sember [4] generalized the natural density by replacing with an arbitrary nonnegative regular matrix . A subset of has -density if exists. Connor [5] and Kolk [6] extended the idea of statistical convergence to -statistical convergence by using the notion of -density.

A sequence is said to be -statistically convergent to if for every . In this case we write . By the symbol we denote the set of all -statistically convergent sequences.

In [7], Edely and Mursaleen generalized these statistical summability methods by defining the statistical -summability and studied its relationship with -statistical convergence.

Let be a nonnegative regular matrix. A sequence is said to be statistically -summable to if for every , ; that is, where . Thus is statistically -summable to if and only if is statistically convergent to . In this case we write . By we denote the set of all statistically -summable sequences. A more general case of statistically -summability is discussed in [8].

Quite recently, Edely [9] defined the concept of -statistical -summability for nonnegative regular matrices and which generalizes all the variants and generalizations of statistical convergence, for example, lacunary statistical convergence [10], -statistical convergence [11], -statistical convergence [6], statistical -summability [7], statistical -summability [12], statistical -summability [13], statistical -summability [14], and so forth.

Let and be two nonnegative regular matrices. A sequence of real numbers is said to be -statistically-summable to if for every , the set has -density zero, thus where In this case we denote by . The set of all -statistically -summable sequences will be denoted by .

Remark 1. (1) If (unit matrix), then is reduced to the set of -statistically convergent sequences which can be further reduced to lacunary statistical convergence and -statistical convergence for particular choice of the matrix .
(2) If matrix, then is reduced to the set of statistically -summable sequences.
(3) If matrix, then is reduced to the set of statistically -summable sequences.
(4) If matrix and are defined by then is reduced to the set of statistically -summable sequences, where is a sequence of nonnegative numbers, such that and
(5) If matrix and are defined by where , then is reduced to the set of statistically -summable sequences.
(6) If a sequence is convergent, then it is -statistically -summable, since converges and has -density zero, but not conversely.
(7) The spaces ,,, and are not comparable, even if .
(8) If a sequence is -summable, then it is -statistically -summable.
(9) If a sequence is bounded and -statistically convergent, then it is -summable and hence statistically -summable ([7], see Theorem 2.1) and -statistically -summable but not conversely.

Example 2. (1) Let us define , , and by Then
Here , , , and , but is -statistically -summable to , since . On the other hand we can see that is -summable and hence is -statistically -summable, -statistically -summable, -statistically convergent, and statistically -summable.

Let denote the linear space of all real-valued functions defined on . Let be the space of all functions continuous on . We know that is a Banach space with norm

We denote by the space of all -periodic functions which is a Banach space with

The classical Korovkin first and second theorems statewhatfollows [15, 16]:

Theorem I. Let be a sequence of positive linear operators from into . Then , for all if and only if   , for , where ,, and .

Theorem II. Let be a sequence of positive linear operators from into . Then ,  for all if and only if   ,  for , where ,, and .

We write for , and we say that is a positive operator if for all .

The following result was studied by Duman [17] which is -statistical analogue of Theorem II.

Theorem A. Let be a nonnegative regular matrix, and let be a sequence of positive linear operators from into . Then for all if and only if

Recently, Karakuş and Demirci [18] proved Theorem II for statistical -summability.

Theorem B. Let be a nonnegative regular matrix, and let be a sequence of positive linear operators from into . Then for all if and only if

Several mathematicians have worked on extending or generalizing the Korovkin's theorems in many ways and to several settings, including function spaces, abstract Banach lattices, Banach algebras, and Banach spaces. This theory is very useful in real analysis, functional analysis, harmonic analysis, measure theory, probability theory, summability theory, and partial differential equations. But the foremost applications are concerned with constructive approximation theory which uses it as a valuable tool. Even today, the development of Korovkin-type approximation theory is far frombeingcomplete. Note that the first and the second theorems of Korovkin are actually equivalent to the algebraic and the trigonometric version, respectively, of the classical Weierstrass approximation theorem [19]. Recently, such type of approximation theorems has been proved by many authors by using the concept of statistical convergence and its variants, for example, [2028]. Further Korovkin type approximation theorems for functions of two variables are proved in [2932]. In [29, 33] authors have used the concept of almost convergence. In this paper, we prove Korovkin second theorem by applying the notion of -statistical -summability. We give here an example to justify that our result is stronger than Theorems II, A, and B. We also study the rate of -statistical -summability of a sequence of positive linear operators defined from into .

#### 2. Main Result

Now, we prove Theorem II for -statistically -summability.

Theorem 3. Let and be nonnegative regular matrices, and let be a sequence of positive linear operators from into . Then for all if and only if

Proof. Since each of 1, , and belongs to , conditions (19) follow immediately from (18). Let the conditions (19) hold and . Let  be a closed subinterval of length of   . Fix . By the continuity of at , it follows that for given there is a number , such that for all whenever . Since is bounded, it follows that for all . For all , it is well known that where . Since the function is -periodic, the inequality (22) holds for .
Now, operating to this inequality, we obtain Now, taking , we get where . Now replace by and then by in (24) on both sides. For a given choose , such that . Define the following sets Then , and so . Therefore, using conditions (19) we get (18).
This completes the proof of the theorem.

#### 3. Rate of -Statistical -Summability

In this section, we study the rate of -statistical -summability of a sequence of positive linear operators defined from into .

Definition 4. Let and be two nonnegative regular matrices. Let be a positive nonincreasing sequence. We say that the sequence is -statistically -summable to the number with the rate if for every , where and as described above. In this case, we write .

As usual we have the following auxiliary result whose proof is standard.

Lemma 5. Let and be two positive nonincreasing sequences. Let and be two sequences, such that and . Then, for any scalar ,, , where .

Now, we recall the notion of modulus of continuity. The modulus of continuity of , denoted by , is defined by

It is well known that

Then prove the following result.

Theorem 6. Let be a sequence of positive linear operators from into . Suppose that(i), (ii), where and  .Then for all , we have where .

Proof. Let and . Using (28), we have Put . Hence we get where . Hence Now, using Definition 4 and Conditions (i) and (ii), we get the desired result.
This completes the proof of the theorem.

#### 4. Example and Concluding Remark

In the following we construct an example of a sequence of positive linear operators satisfying the conditions of Theorem 3 but does not satisfy the conditions of Theorems II, A, and B.

For any , denote by the th partial sum of the Fourier series of ; that is,

For any , write

A standard calculation gives that for every where The sequence is a positive kernel which is called the Fejér kernel, and the corresponding operators ,, are called the Fejér convolution operators. We have

Note that the Theorems II, A, and B hold for the sequence . In fact, we have for every ,

Let , and be defined as in Example 2. Let be defined by Then is not statistically convergent, not -statistically convergent, and not statistically -summable, but it is -statistically summable to . Since is -statistically -summable to , it is easy to see that the operator satisfies the conditions (19), and hence Theorem 3 holds. But on the other hand, Theorems II, A, and B do not hold for our operator defined by (39), since (and so ) is not statistically convergent, not -statistically convergent, and not statistically -summable.

Hence our Theorem 3 is stronger than all the above three theorems.

#### Acknowledgments

This joint work was done when the first author visited University Putra Malaysia as a visiting scientist during August 27–September 25, 2012. The author is very grateful to the administration of UPM for providing him local hospitalities.