Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ

We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter λ ∈ [−1, 1] and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type λ-Bernstein operators and study their approximation behaviors.


Introduction
A famous mathematician Bernstein [1] constructed polynomials nowadays called Bernstein polynomials, which are familiar and widely investigated polynomials in theory of approximation.Bernstein gave a simple and very elegant way to obtain Weierstrass approximation theorem with the help of his newly constructed polynomials.For any continuous function f (x) defined on C[0, 1], Bernstein polynomials of order n are given by where the Bernstein basis functions b n,i (x) are defined by b n,i (x) = n i x i (1 − x) n−i (i = 0, . . ., n).
Stancu [2] presented a generalization of Bernstein polynomials with the help of two parameters α and β such that 0 ≤ α ≤ β, as follows: If we take both the parameters α = β = 0, then we get the classical Bernstein polynomials.The operators defined by ( 2) are called Bernstein-Stancu operators.For some recent work, we refer to [3][4][5][6].
In the recent past, Cai et al. [7] presented a new construction of Bernstein operators with the help of Bézier bases with shape parameter λ and called it λ-Bernstein operators, which are defined by where bn,i (λ; x) are Bézier bases with shape parameter λ (see [8]), defined by bn,0 (λ; x) = b n,0 (x) in this case λ ∈ [−1, 1] and b n,i (x) are the Bernstein basis functions.By taking the above operators into account, they established various approximation results, namely, Korovkin-and Voronovskaja-type theorems, rate of convergence via Lipschitz continuous functions, local approximation and other related results.In the same year, Cai [9] generalized λ-Bernstein operators by constructing the Kantorovich-type λ-Bernstein operators, as well as its Bézier variant, and studied several approximation results.Later, various approximation properties and asymptotic type results of the Kantorovich-type λ-Bernstein operators have been studied by Acu et al. [10].Very recently, Özger [11] obtained statistical approximation for λ-Bernstein operators including a Voronovskaja-type theorem in statistical sense.In the same article, he also constructed bivariate λ-Bernstein operators and studied their approximation properties.The Bernstein operators are some of the most studied positive linear operators which were modified by many authors, and we are mentioning some of them and other related work [12][13][14][15][16][17][18][19][20][21][22][23].
The rest of the paper is organized as follows: In Section 2, we calculate the moments of (5) and prove global approximation formula in terms of Ditzian-Totik uniform modulus of smoothness of first and second order.The local direct estimate of the rate of convergence by Lipschitz-type function involving two parameters for λ-Bernstein-Stancu operators is investigated.In Section 3, we establish quantitative Voronovskaja-type theorem for our operators.The final section of the paper is devoted to study the bivariate case of λ-Bernstein-Stancu operators .

Some Auxiliary Lemmas and Approximation by Stancu-Type λ-Bernstein Operators
In this section, we first prove some lemma which will be used to study the approximation results of (5).Lemma 1.For x ∈ [0, 1], the moments of Stancu-type λ-Bernstein operators are given as: Proof.Using the definition of operators ( 5) and Bézier-Bernstein bases bn,i (λ; x) (4), we write where .
Again, by using the following identity; together with (4) and (5), we can write where .

Corollary 1.
The following relations hold: Corollary 2. The following identities hold: We obtain the uniform convergence of operators B λ n,α,β ( f ; x) by applying well-known Bohman-Korovkin-Popoviciu theorem.
Theorem 1.Let C[0, 1] denote the space of all real-valued continuous functions on [0, 1] endowed with the supremum norm.Then as stated in Bohman-Korovkin-Popoviciu theorem.We have the following relations by Lemma 1: It is easy to show and hence Recall that the first and second order Ditzian-Totik uniform modulus of smoothness are given by respectively, where φ is an admissible step-weight function on [24]).Let be the corresponding K-functional, where In this case, g ∈ AC[0, 1] means that g is absolutely continuous on [0, 1].It is known by [25] that there exists an absolute constant C > 0, such that We are now ready to obtain global approximation theorem.
We obtain the following relations by applying the Taylor's formula: By using the definition of K-functional together with (6) and the inequalities ( 9) and ( 10), we have Also, by first order Ditzian-Totik uniform modulus of smoothness, we have Therefore, the following inequalities hold: which completes the proof.
In order to obtain next result, we first recall some concepts and results concerning modulus of continuity and Peetre's K-functional.For δ > 0, the modulus of continuity w It is also well known that, for any δ > 0 and each For f ∈ C[0, 1], the second-order modulus of smoothness is given by and the corresponding Peetre's K-functional [26] is where It is well-known that the inequality holds in which the absolute constant C > 0 is independent of δ and f (see [25]).
We are now ready to establish a direct local approximation theorem for operators B λ n,α,β ( f ; x) via second order modulus of smoothness and usual modulus of continuity.
Proof.Consider the operators Bλ n,α,β ( f ; x) as defined in Theorem 2. Assume that t, x ∈ [0, 1] and g ∈ W 2 [0, 1].The following equality yields by Taylor's expansion formula: If we apply Bλ n,α,β (•; x) to both sides of ( 13) and keeping in mind these operators preserve constants and linear functions, we obtain Therefore, With the help of (7), one obtains Now, for f ∈ C[0, 1] and g ∈ W 2 [0, 1], using ( 7) and ( 14), we get Finally, by assuming the infimum on the right-hand side of the above inequality over all g ∈ W 2 [0, 1] togrther with inequality (12), we obtain which completes the proof.
In the following theorem, we obtain a local direct estimate of the rate of convergence via Lipschitz-type function involving two parameters for the operators B λ n,α,β .Before proceeding further, let us recall that for k 1 ≥ 0, k 2 > 0, where η ∈ (0, 1] and M is a positive constant (see [27]).
Proof.We first write the following equality by Taylor's expansion theorem of function f (x) in C B [0, 1]: where r x (t) is Peano form of the remainder, r x ∈ C[0, 1] and r x (t) → 0 as t → x.Applying the operators B λ n,α,β (•; x) to identity (17), we have Using Cauchy-Schwarz inequality, we have We observe that lim n B λ n,α,β (r 2 x (t); x) = 0 and hence The result follows immediately by applying the Corollaries 1 and 2.
Proof.Consider the following equality Applying B λ n,α,β (•; x) to both sides of (20), we obtain The quantity in the right hand side of ( 21) can be estimated as where g ∈ W φ [0, 1].There exists C > 0 such that for sufficiently large n.By taking ( 21)-( 23) into our account and using Cauchy-Schwarz inequality, we have Finally, by taking infimum over all g ∈ W φ [0, 1], this last inequality leads us to the assertion (19) of Theorem 7.
As an immediate consequence of Theorem 7, we have the following result.

4.
The Bivariate Case of the Operators B λ n,α,β ( f ; x) We construct bivariate version of Stancu-type λ-Bernstein operators defined which was defined in the first section of this manuscript as (5) and study their approximation properties.
Theorem 8. Let e ij (x, y) = x i y j , where 0 ≤ i + j ≤ 2.Then, the sequence B λ,α,β n,m ( f ; x, y) of operators converges uniformly to f on I for each f ∈ C (I). Keeping in mind the above conditions and Korovkin type theorem established by Volkov [30], we obtain lim m,n→∞ B λ,α,β n,m ( f ; x, y) = f converges uniformly.Now, we compute the rate of convergence of operators (24) by means of the modulus of continuity.Recall that the modulus of continuity for bivariate case is defined as Peetre's K-functional is given by for δ > 0, where C 2 (I ab ) is the space of functions of f such that f , ∂ j f ∂x j and ∂ j f ∂y j (j = 1, 2) in C(I ab ) [26].We now give an estimate of the rates of convergence of operators B λ,α,β n,m ( f ; x, y).
for all x ∈ I, where (s − x) .
Proof.By using the definition of partial modulus of continuity and Cauchy-Schwartz inequality, we have Finally, by choosing ν n (α, β, λ; x) and ν m (α, β, λ; y) as defined in Theorem 9, we obtain desired result.
We recall that the Lipschitz class Lip M ( β 1 , β 2 ) for the bivariate is given by Proof.We have . Then, by applying the Hölder's inequality for Proof.We have
) and for every (s, t), (x, y)∈ I ab = [0, a] × [0, b].The partial moduli of continuity with respect to x and y are defined by