On Some Statistical Approximation by ( p , q )-Bleimann , Butzer and Hahn Operators

In this article, we propose a different generalization of (p, q)-BBH operators and carry statistical approximation properties of the introduced operators towards a function which has to be approximated where (p, q)-integers contains symmetric property. We establish a Korovkin approximation theorem in the statistical sense and obtain the statistical rates of convergence. Furthermore, we also introduce a bivariate extension of proposed operators and carry many statistical approximation results. The extra parameter p plays an important role to symmetrize the q-BBH operators.


Introduction
The q-analog of Bleiman, Butzer and Hahn operators (BBH) [1] is defined by: where q n (x) = ∏ n−1 k=0 (1 + q s x).For q = 1, the sequence of q-BBH operators (1) reduces to the classical BBH-operators [2] in which authors investigated pointwise convergence properties of the BBH-operators in a compact sub-interval of R + .
Let H ω denote the space of all real-valued functions f defined on the semi-axis R + [3], where ω is the usual modulus of continuity satisfying Gadjiev and Çakar [3] established the Korovkin type theorem which gives the convergence for the sequence of linear positive operators (LPO) to the functions in H ω .Now, we recollect the following theorem: Theorem 1 ([3]).Let {A n } be the sequence of LPOs from H ω into C B (R + ) such that (p, q)-calculus, also called post-quantum calculus, is a generalization of q-calculus which has lots of applications in quantum physics.In approximation theory, the very first (p, q)-type generalization of Bernstein polynomials was introduced by Mursaleen et al. [4] using (p, q)-calculus and improved the said operators (see Erratum [4]).The theory of semigroups of the linear operators is used in order to prove the existence and uniqueness of a weak solutions of boundary value problems in thermoelasticity of dipolar bodies (see [5,6]).
[n] p,q stands for (p, q)-integers defined as a n−j b j x n−j y j , and the binomial coefficients in (p, q)-calculus are given by By easy computation, we have the relation given below: Authors suggest the readers [16][17][18][19].
The (p, q)-analogue of BBH operators was introduced by Mursaleen et al. in [20] as follows: where x ≥ 0, 0 < q < p ≤ 1, p,q n (x) = ∏ n−1 s=0 (p s + q s x) and function f is defined on the semi axis R + .If we put p = 1, we get the q-BBH operators (1).In that paper, authors established different approximation properties of the sequence of operators (3).
Mursaleen and Nasiruzzaman constructed bivariate (p, q)-BBH operators [21] and studied many nice properties based on that sequence of operators and also given some generalization of that sequence of bivariate operators introducing one more parameter γ in the operators.
The statistical convergence is another notion of convergence, which was introduced by Fast [22] nearly fifty years ago and now this is a very active area of research.The statistical limit of a sequence is an extension of the idea of limit of sequence in an ordinary sense.The natural density of K ⊂ N is defined as: {k ≤ n : k ∈ K} whenever the limit exists (see [23,24]).The sequence x = (x k ) is said to be statistically convergent to a number L means if, for every > 0, and it is denoted by st − lim k x k = L.It can be easily seen that every convergent sequence is statistically convergent but not conversely.Now, we will state some preliminary results on positive linear operators: ).If L is an operator, linear and positive, then, for every x ∈ X, we have Proposition 2 ( [25]).(Hölder's inequality for LPOs).Let L : X → Y be an operator, linear and positive, and let 1/p + 1/q = 1, where p, q > 1 are real numbers.Then, for every f Remark 1 ([25]).A particular case of Proposition 2 is the Cauchy-Schwarz's inequality for LPOs, which is obtained from Hölder's inequality for p = q = 2 as: We have organized the rest of the paper as follows.In Section 2, we have constructed (p, q)-BBH operators and calculated some auxiliary results.In Sections 3 and 4, Korovkin type results and rate of convergence are established in statistical sense, respectively.Section 5 is devoted to the construction of the bivariate (p, q)-BBH operators.In Section 6, we have computed rate of statistical convergence for the bivariate (p, q)-BBH operators.

Construction of Operators and Moment Estimation
Ersan and Do gru [26] introduced a generalization of (1) and studied different statistical approximation properties of the operators towards a function f which has to be approximated.Inspired with the work of Ersan and Do gru [26], we construct a (p, q)-analogue generalization of the sequence of operators defined in [26] or, on the other hand, we generalize the operators introduced in [20] as follows: ∀x ≥ 0, 0 < q < p ≤ 1, let us define a sequence of (p, q)-BBH operators as follows: where It is easy to verify that, if p = q = 1, the operators turn into the classical BBH operators.The sequence of operators ( 7) is of course more generalized than (1), and it is more flexible than (1).
We need the following lemma to our main result: Let the sequence of operators be given by (7).Then, for any x ≥ 0, 0 < q < p ≤ 1.
Proof.The proof is obvious with the help of the relation ( 8), so we skip the proof.
By using (8), the result can be easily obtained.
Lemma 3. Let the sequence of operators be given by (7).Then, B p,q n t 1 + t for any x ≥ 0, 0 < q < p ≤ 1.
Proof.It is easy to verify that With the help of ( 12), we can have Now, using (8), we can get the desired result.

Korovkin Type Statistical Approximation Properties
In this section, we obtain the Korovkin type statistical approximation theorem for our sequence of operators (7).Let us give the following theorem: Let us take p = (p n ) and q = (q n ) such that Theorem 4. Let B p,q n ( f ; x) be the sequence of operators (7) and the sequences p = (p n ) and q = (q n ) satisfy the assumption (13) for 0 < q n < p n ≤ 1.Then, for any function Proof.For ν = 0 and using (9), we can have By (13), the following can be easily verified, which is For ν = 1 and using (10), we get For a given > 0, let us define the following sets: It is easily perceived that U ⊂ U , so we can write On using (13), it is clear that Thus, Lastly, for ν = 2 and using (11), we obtain Using [n + 1] p n ,q n = p n [n] p n ,q n + q n n , the following can be easily justified that Substituting it in ( 14), we can have , and , then by ( 13), we have For any given > 0, now we define four sets as follows: It is clear that the right-hand side of the above inequality is zero by (15); then, Hence, the proof is completed.

Rates of Statistical Convergence
This section is devoted to find rates of statistical convergence of operators (7).The modulus of continuity for the space of functions f ∈ H ω [1] is defined by where ω( f ; δ) satisfies the following conditions: Theorem 5. Let p = (p n ) and q = (q n ) be the sequences satisfying (13) and 0 < q n < p n ≤ 1, we have where Proof.

|B
p n ,q n n By using the Cauchy-Schwarz inequality (see (6)) and using ( 9)-( 11), we have Thus, it is obvious that, by choosing δ n as in (16), the theorem is proved.
Notice that, by conditions in (13), st − lim n = 0.Then, we have This provides us the pointwise rate of statistical convergence of the sequence of operators B p n ,q n n ( f ; x) to f (x).Now, we will contribute an estimate related to the rate of approximation by means of Lipschitz type maximal functions.
Lenze [27] introduced a Lipschitz type maximal function as The Lipschitz type maximal function space on E ⊂ R + is defined in [1] as follows: where function f is bounded and continuous on R + , 0 < α ≤ 1 and M is a positive constant. where Proof.A similar technique used in Theorem 7 in [26] will be taken to provide the proof.Letting x ≥ 0, (x, Since B p n ,q n n ( f ; x) is a linear and positive operator, f ∈ W α,E , using the previous inequality, we have Consequently, we obtain Using the Hölder's inequality (see (5)) with p = 2 α and q = 2 2−α and using ( 9)-( 11), we have If the above result is substituted in (17), we will get our desired result.Hence, the theorem is proved.

Corollary 1. If B
p n ,q n n ( f ; x) is defined by (7) and take E = R + implies d(x, E) = 0, then a special case of Theorem 6 can be obtained as the following result: , where ρ n (x) is the same as in Theorem 6.

Construction of the Bivariate Operators
In this section, we define a bivariate version of operators (7) and study their approximation properties.
For R 2 + = [0, ∞) × [0, ∞), f : R 2 + → R and 0 < q n 1 , q n 2 < p n 1 , p n 2 ≤ 1, let us define the bivariate case of the operators (7) as follows: where ζ Details of the modulus of continuity for the bivariate case can be found in [28].Now, we will investigate Korovkin type approximation properties by using the following test functions:
In the end, we will present the rates of statistical convergence of the bivariate operators (18) by means of Lipschitz type maximal functions.