A -Statistical Convergence Properties of Kantorovich Type λ -Bernstein Operators Via ( p , q )-Calculus

: In the present paper, Kantorovich type λ -Bernstein operators via ( p , q )-calculus are constructed, and the ﬁrst and second moments and central moments of these operators are estimated in order to achieve our main results. An A -statistical convergence theorem and the rate of A -statistical convergence theorems are obtained according to some analysis methods and the deﬁnitions of A -statistical convergence, the rate of A -statistical convergence and modulus of smoothness.


Introduction
As we know, one of the simplest and most elegant ways to prove the famous Weierstrass Approximation Theorem was given by S. N. Bernstein [1] in 1912 by constructing a sequence of polynomials which were defined as follows, Then, there are many papers mention about the approximation properties of (p, q)-type positive linear operators, such as .
[n] p,q ! and n k p,q are defined as follows: [n] p,q ! = [n] p,q [n − 1] p,q ... [1] p,q , n = 1, 2, · · · ; 1, The (p, q)-power basis (x ⊕ t) n p,q and (x t) n p,q are defined by We also give the fundamental theorem of (p, q)-calculus, say, if F(x) is an anti-derivative of f (x) and F(x) is continuous at the infinite series here converges.
The main goal of the present work is to study the rate of A-statistical convergence of Kantorovich type λ-Bernstein operators based on (p, q)-integers by means of modulus of continuity. The rest of this paper are mainly organized as follows: in Section 2, some moments and central moments of K λ n,p,q ( f ; x) are estimated; in Section 3, we prove K λ n,p,q ( f ; x) is A-statistically convergent to f (x) and investigate the rate of A-statistical convergence by means of the first and second modulus continuity.

Some Preliminary Results
In the sequel, consider sequences of functions e i (x) = x i (i = 0, 1, 2), and φ j (t, x) = (t − x) j (j = 1, 2), x, t ∈ [0, 1]. Before we give our main theorems, we need the following lemmas. Lemma 1. The following statements are true: Proof. By the fundamental theorem of (p, q)-calculus given in Section 1, we have Similarly, Lemma 1 is proved.

Proof. By Equations
Similarly, by Lemma 2, we have Next, we will discuss in two cases:  .

A-Statistical Convergence Properties
Let C[0, 1] be the space of all real-valued continuous bounded functions f on [0, 1], endowed with the norm || f || C[0,1] = sup x∈[0,1] | f (x)|. In this section, we will give some A-statistical convergence properties for positive linear operators K λ n,p,q ( f ; x) by the following definition of A-statistical convergence and the first and second modulus of continuity. [29]) For a given non-negative infinite summability matrix A = (a nk ), n, k ∈ N, A-transform of x denoted by Ax := {(Ax) n } is defined as (Ax) n = ∑ ∞ k=0 a nk x k provided the series converges for each n. We say that A is regular if lim n (Ax) n = L whenever lim x = 1. Assume that A is non-negative regular summability matrix, a sequence x = {x k } is called A-statistically convergent to L provided that for every > 0, lim n ∑ k:|x k −L|≥ a nk = 0. We denote this limit by st A − lim x = L.

Definition 1. (See
As we know, A-statistical convergence becomes ordinary statistical convergence when A = (C 1 ), the Cesaro matrix of order one, and it becomes classical convergence when A = I, the identity matrix. There is also a conclusion, every convergent sequence is statistically convergent to the same limit but not conversely.
We need the following Korovkin theorem via the conception of A-statistical convergence to prove our main results.

Theorem 2.
(See [29]) Let A = (a nk ) be a non-negative regular summability matrix and L n ( f ; x) be positive linear operators over C [a, b]. Then the following two statements are equivalent: Consider sequences p := {p n }, q := {q n } for 0 < q n < p n ≤ 1 satisfying the following conditions We now give main results related to statistical convergence of operators in Equation (5).
Theorem 3. Let A = (a jn ) be a weighted non-negative regular summability matrix for n, k ∈ N, Proof. According to Theorem 2, it is sufficient fo satisfy From Lemma 2, it is clear that Equation (20) is true for i = 0. For i = 1, by Equation (15), we have Given > 0, we define the following sets: Letting j → ∞ in Equation (21), using Equation (18) and We obtain st A − lim n K λ n,p,q (e 2 ) − e 2 C[0,1] = 0. Therefore, Equation (20) is proved, which yields the result of Theorem 3.
We now need the following definitions to estimate the rate of A-statistical convergence of K λ n,p,q ( f ; x). In this case we write where δ > 0 and C 2 [0,1] = g ∈ C[0, 1] : g , g ∈ C[0, 1] . The second order modulus of smoothness of f ∈ C[0, 1] is defined by There exist an absolute constant C > 0 such that K 2 ( f ; δ) ≤ Cω 2 f ; √ δ . We also denote the usual of modulus of continuity by Theorem 6. Let A = (a jn ) be a non-negative regular summability matrix.
Assume that where Φ(p, q; n) is defined in Equation (16). Then for f ∈ C[0, 1], we have Proof. Since Applying K λ n,p,q ( f ; x) to both ends and using Cauchy-Schwarz inequality, we have Letting δ = K λ n,p,q (φ 2 (t, x)), we get where Φ(p, q; n) is defined in Equation (16). Taking supremum over [0, 1] on both sides, we obtain For a given > 0, consider following sets Obviously, we have V ⊆ V 1 and we also can obtain Thus, let j → ∞, by hypothesis we are led to the fact that K λ n,p,q ( f ) − f Theorem 6 is proved.

Conclusions
In this paper, we introduced a kind of Kantorovich type λ-Bernstein operators K λ n,p,q ( f ; x) via (p, q)-calculus, we estimated the moments and central moments and used these results to obtain an A-statistical convergence theorem and the rate of A-statistical convergence of K λ n,p,q ( f ; x) to f (x). In the future research work, we will continue to investigate some approximation properties of Durrmeyer type λ-Bernstein operators via (p, q)-calculus.
Author Contributions: All authors contribute equally to this article. All authors have read and agreed to the published version of the manuscript.