Modified (p,q)-Bernstein-Schurer operators and their approximation properties

In this paper, we introduce modified (p, q)-Bernstein–Schurer operators and discuss their statistical approximation properties based on Korovkins type approximation theorem. We compute the rate of convergence and also prove a Voronovskaja-type theorem.


Introduction and preliminaries
In Lupaş (1987) introduced the first q-analogue of the classical Bernstein operators and investigated its approximating and shape-preserving properties. Another q-generalization of the classical Bernstein polynomial is due to Phillips (1997). Several generalizations of well-known positive linear operators based on q-integers were introduced and their approximation properties have been studied by several researchers.

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In this paper, we have modified the (p, q)-Bernstein-Schurer operators and discussed their statistical approximation properties based on Korovkin's type approximation theorem. We have also established the rate of convergence of these operators using the modulus of continuity. Furthermore, we have proved a Voronovskaja-type theorem. One of its advantages of using the extra parameter p has been mentioned in Mursaleen, Faisal Khan and Asif Khan (2016) to study (p, q)approximation by Lorentz operators in compact disk. Another nice application has been given by Khan, Lobiyal and Kilicman (2015) and  in computer-aided geometric design and applied these Bernstein bases for construction of (p, q)-Bézier curves and surfaces based on (p, q)integers. Mursaleen, Nasiuzzaman, and Nurgali (2015) and Mursaleen and Nasiruzzaman (2015). One of its advantages of using the extra parameter p has been mentioned in  to study (p, q)-approximation by Lorentz operators in compact disk. Another nice application has been given by  and  in computer-aided geometric design and applied these Bernstein bases for construction of (p, q)-Bézier curves and surfaces based on (p, q)-integers.
The (p, q)-integer was introduced to generalize or unify several forms of q-oscillator algebras well known in the Physics literature related to the representation theory of single-parameter quantum algebras. The (p, q)-integer is defined by The (p, q)-binomial coefficients are defined by In Schurer (1962) introduced and studied the operators C m,d :C[0, d + 1] → CC[0, 1]) defined for any m ∈ ℕ and d be fixed in ℕ and any function f ∈ C[0, d + 1] as follows: In Muraru (2011) constructed the q-Bernstein-Schurer operators defined by  introduced the generalized (p, q)-analogue of Bernstein-Schurer operators as follows: x ∈ [0, 1].

Construction of operators
We consider 0 < q < p ≤ 1 and for any m ∈ ℕ, (iv)

Statistical approximation
First, we recall the concept of statistical convergence for sequences of real numbers which were introduced by Fast (1951) and further studied by many others. Let K ⊆ ℕ and K n = {j ≤ n:j ∈ K}. The natural density of K is defined by (K) = lim n→∞ 1 n |K n | if the limit exists, where |K n | denotes the cardinality of the set K n . A sequence x = (x k ) of real numbers is said to be statistically convergent to L, provided that for every > 0, the set {j ∈ ℕ:|x j − L| ≥ } has natural density zero, that is for each > 0, In this case, we write st − lim n x n = L. Note that every convergent sequence is statistically convergent but not conversely. For example, let u = (u m ) be defined by then, st − lim u m = 0, but u is not convergent. Recently, the idea of statistical convergence has been used in proving some approximation theorems by various authors and it was found that the statistical versions are stronger than the classical ones. Authors have used many types of classical operators and test functions to study the Korovkin-type approximation theorems which further motivate continuation of this study. After the paper of Gadjiev and Orhan (2002), different types of summability methods have been deployed in approximation process, for example, Mursaleen, Khan, Srivastava, and Nisar (2013), Mursaleen and Kilicman (2013). In this section, we obtain the Korovkintype weighted statistical approximation properties for these operators. for any x, y ∈ [0, d + 1]. For q ∈ (0, 1) and p ∈ (q, 1] it is obvious that lim m→∞ [m] p,q = 1 p−q . In order to reach to convergent result of the operator L p,q m,d , we take a sequence q m ∈ (0, 1), p m ∈ (q, 1] such that

Rates of convergence
We will estimate the rate of convergence in terms of modulus of continuity. Let f ∈ C [0, b], and the modulus of continuity of f denoted by (f , ) gives the maximum oscillation of f in any interval of length not exceeding > 0 and it is given by the relation It is known that lim →0+ (f , ) = 0 for f ∈ C[0, b] and for any > 0, one has where Proof Using the Cauchy inequality and lemma (2.1), we have Hence, the desired result is obtained. ✷ . Now from (5.1), (5.2) and Lemma 2.2 (ii), we get Now we compute the following: