Approximation properties of Chlodowsky variant of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}(p,q) Bernstein-Stancu-Schurer operators

In the present paper, we introduce the Chlodowsky variant of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}(p,q) Bernstein-Stancu-Schurer operators which is a generalization of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}(p,q) Bernstein-Stancu-Schurer operators. We also discuss its Korovkin-type approximation properties and rate of convergence.


Introduction and preliminaries
Later various generalizations of these operators were discovered. It has been proved as a powerful tool for numerical analysis, computer aided geometric design and solutions of differential equations. In last two decades, the applications of q-calculus has played an important role in the area of approximation theory, number theory and theoretical physics. In , Lupaş For the first few moments, we get the following lemma.

Construction of the operators
Considering the revised form of (p, q) analogue of Bernstein operators [], we construct the Chlodowsky variant of (p, q) Bernstein-Stancu-Schurer operators as where n ∈ N, m, α, β ∈ N  , with α β ≈ ,  ≤ x ≤ b n ,  < q < p ≤  and b n is an increasing sequence of positive terms with the properties b n → ∞ and b n [n] p,q →  as n → ∞. Evidently, C (α,β) n,m is a linear and positive operator. Consider the case if p, q →  and m =  in (.), then it will reduce to the Stancu-Chlodowsky polynomials [].
Let us assume the number n + m = n m , we will use this notation throughout in this paper. Next, we have obtained the following lemma using simple calculations.
Using the linear property of operators, we have Hence, we get (iv). Similar calculations give Substituting the results of (i), (ii) and (iii), we prove the result (v).

Lemma  For every fixed
We can conclude the last inequality using the following statements: Remark  As a result of Lemma  and , we have

Results and discussion
In this paper we have constructed and investigated a Chlodowsky variant of (p, q) Bernstein-Stancu-Schurer operator. We have showed that our modified operators have a better error estimation than the classical ones. We have also obtained some approximation results with the help of the well-known Korovkin theorem and the weighted Korovkin theorem for these operators. Furthermore, we studied convergence properties in terms of the modulus of continuity for functions in Lipschitz class. Next we have also obtained the Voronovskaja-type result for these operators.

Korovkin-type approximation theorem
Assume C ρ is the space of all continuous functions f such that and ρ(x) is the weight function.
Then C ρ is a Banach space with the norm is finite}. The subsequent Theorem  is a Korovkin approximation theorem in weighted space.

Theorem  (See [])
There exists a sequence of positive linear operators U n , acting from C  ρ to C  ρ , satisfying the conditions () lim n→∞ U n (; ·) - ρ = , is a continuous and increasing function on (-∞, ∞) such that lim x→±∞ φ(x) = ±∞ and ρ(x) =  + φ  , and there exists a function f * ∈ C  ρ for which Consider the weight function ρ(x) =  + x  and operator (see []) For f ∈ C +x  , we have Now, using Lemma  we will obtain n,m (f ; ·, p, q) is bounded operator, henceforth a continuous operator too. Since ' An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. ' Now, consider the sequences (p n ) and (q n ) for  < q n < p n ≤  satisfying provided that p := (p) n , q := (q) n with  < q n < p n ≤  satisfying (.) and lim n→∞ b n [n] pn,qn = .
Proof Using the results of Theorem  and Lemma (i), (ii) and (iii), we will obtain the following assessments, respectively: whenever n → ∞.
Since the weight function is invariant w.r.t. positive and negative values of x, and conditions (.)-(.) are true for all t ∈ R, we can use Theorem  and get the desired result (.), which implies that the operator sequence C α,β n,m converges uniformly to any continuous function in weighted space C  +x  for x ∈ [, b n ].

Theorem  Assuming c as a positive and real number independent of n and f as a continuous function which vanishes on
where x ∈ [, b n ] and δ = δ( ) are independent of n. Operating with the operator (.) on both sides, we can conclude by using Lemma  and Remark , [n] pn,qn =  as n → ∞, we have the desired result.

Rate of convergence
We will find the rate of convergence for functions in the Lipschitz class Lip M (γ ) ( < γ ≤ ). Assume that C B [, ∞) denotes the space of bounded continuous functions on [, ∞).
Proof Since f ∈ Lip M (γ ), and the operator C α,β n,m (f ; x, p, q) is linear and monotone, Using Hölder's inequality with the values p =  γ and q =  -γ , we get In order to obtain rate of convergence in terms of modulus of continuity ω(f ; δ), we assume that, for any f ∈ C B [, ∞) and x ≥ , the modulus of continuity of f is given by Thus it implies for any δ >  where ω(f ; ·) is the modulus of continuity of f and λ n,p,q (x) is the same as in Theorem .
Proof Using the triangular inequality, we get Now using (.) and Hölder's inequality, we get Now choosing δ = λ n,p,q (x) as in Theorem , we have Next we calculate the rate of convergence in terms of the modulus of continuity of the derivative of a function.

Voronovskaja-type result
Now, we prove a Voronovskaja-type approximation theorem with the help of the C [n] p n ,q n b n C (α,β) n,m (tx)  ; x, p n , q n = ax, where a ∈ (, ).
Proof We shall prove only (.) because the proof of (.) is similar. Let x ∈ [, E]. Then, by Lemma (), we obtain, for all n ∈ N, Now by taking the limit as n → ∞ in (.), we obtain which completes the proof.
In a similar way to Lemma  one can deduce the following lemma.
Lemma  Let (p n ) and (q n ) be two sequences satisfying (.) and x ∈ [, E] where E ∈ R + .
There is a positive constants M  (x) depending only on x such that Proof Using the Taylor formula for f ∈ C  +x  , we have

Example
With

Conclusion
A better approximation of complex functions over the required interval [, b n ] can be attained using the Chlodowsky variant of the (p, q) Bernstein-Stancu-Schurer operator for choosing suitable values of the sequence b n and n compared to classical operators over the fixed interval [, ].