Some approximation results on Bernstein-Schurer operators defined by (p,q)-integers (Revised)

In the present article, we have given a corrigendum to our paper"Some approximation results on Bernstein-Schurer operators defined by (p,q)-integers"published in Journal of In- equalities and Applications (2015) 2015:249.


Introduction and Preliminaries
In 1912, S.N Bernstein [4] introduced the following sequence of operators B n : C[0, 1] → C[0, 1] defined for any n ∈ N and for any f ∈ C[0, 1] such as In 1987, Lupas [8] introduced the q-Bernstein operators by applying the idea of q-integers, and in 1997 another generalization of these operators introduced by Philip [19]. Later on, many authors introduced q-generalization of various operators and investigated several approximation properties.
The (p, q)-integer was introduced in order to generalize or unify several forms of q-oscillator algebras well known in the earlier physics literature related to the representation theory of single parameter quantum algebras [5]. The (p, q)-integer [n] p,q is defined by [n] p,q = p n − q n p − q , n = 0, 1, 2, · · · , 0 < q < p ≤ 1.
In 1962, Schurer [21] introduced and studied the operators S m,ℓ : C[0, ℓ + 1] → C[0, 1] defined for any m ∈ N and ℓ be fixed in N and any function f ∈ C[0, ℓ + 1] as follows For any m ∈ and f ∈ C[0, ℓ + 1], p is fixed, then q− analogue of Bernstein-Schurer operators in [11] defined as follows Recently, Mursaleen et al [14] applied (p, q)-calculus in approximation theory and introduced first (p, q)-analogue of Bernstein operators. They have also introduced and studied approximation properties of (p, q)-analogue of Bernstein-Stancu operators in [15] and (p, q)analogue of Bernstein-Kantorovich Operators in [16] as well as approximation by (p, q)-Lorentz polynomials on a compact disk in [17].
Our aim is to introduce (p, q)-analogue of Bernstein-Schurer operators. We investigate the approximation properties of this class and we estimate the rate of convergence and some theorem by using the modulus of continuity. We study the approximation properties based on Korovkin's type approximation theorem and also establish the some direct theorem.
Lemma 2.2. Let B p,q m,ℓ (.; .) be given by lemma 2.1, then for any x ∈ [0, 1] and 0 < q < p ≤ 1 we have the following identities On the convergence of (p, q)-Bernstein-Schurer operators Let f ∈ C[0, γ], and the modulus of continuity of f denoted by ω(f, δ) gives the maximum oscillation of f in any interval of length not exceeding δ > 0 and it is given by the relation It is known that lim δ→0+ ω(f, δ) = 0 for f ∈ C[0, γ] and for any δ > 0 one has For q ∈ (0, 1) and p ∈ (q, 1] obviously have lim m→∞ [m] p,q = 1 p−q . In order to reach to the convergence results of the operator B p,q m,ℓ , we take a sequence q m ∈ (0, 1) and p m ∈ (q m , 1] such that lim m→∞ p m = 1 and lim m→∞ q m = 1, so we get lim m→∞ [m] pm,qm = ∞.
Proof. The proof is based on the well known Korovkin theorem regarding the convergence of a sequence of linear and positive operators, so it is enough to prove the conditions By taking the simple calculation we get Proof.
By using the Cauchy inequality and lemma 2.1 we have Hence we obtain the desired result by choosing δ = δ m .

Direct Theorems on (p, q)-Bernstein-Schurer operators
The Peetre's K-functional is defined by Then there exits a positive constant C > 0 such that K 2 (f, δ) ≤ Cω 2 (f, δ 1 2 ), δ > 0, where the second order modulus of continuity is given by Theorem 4.1. Let f ∈ C[0, ℓ + 1], g ′ ∈ C[0, ℓ + 1] and satisfying 0 < q < p ≤ 1. Then for all n ∈ N there exits a constant C > 0 such that Proof. Let g ∈ W 2 , then from the Taylor's expansion, we get Now by lemma 2.1, we have On the other hand we have Since we know the relation Now taking the infimum on the right hand side over all g ∈ W 2 , we get In the view of the property of K-functional, we get This completes the proof.
Proof. From the Taylors formula we have where r(t, x) is the remainder term and lim t→x r(t, x) = 0, therefore we have Since r 2 (x, x) = 0, and r 2 (t, x) ∈ C[0, ℓ + 1], then for from the Theorem (3.1) we have This completes the proof. Now we give the rate of convergence of the operators B p,q m,ℓ (f ; x) in terms of the elements of the usual Lipschitz class Lip M (ν).