APPROXIMATION PROPERTIES OF λ -BERNSTEIN-KANTOROVICH-STANCU OPERATORS

. The goal of this paper is to construct a new type of Bernstein operators depending on the shape parameter λ ∈ [ − 1 , 1]. For these new type operators a uniform convergence result is presented. Furthermore, order of approximation in the sense of local approximation is investigated and Voronovskaja type theorem is proved. Lastly, some graphical results are given to show the rate of convergence of constructed operators to a given function f .


Introduction
In 1912, the Russian mathematician S. N. Bernstein gave one of the most known proof of the Weierstrass's theorem [17], in [5], by defining the following operators; B n : where In 1987, Lupaş [12] introduced the first q-analogue of Bernstein operators and investigated its approximating and shape-preserving properties.In 1996, Phillips [14] proposed another q-variant of the classical Bernstein operators, the so-called Phillips q-Bernstein operators.We have to remind that the q-variants of Bernstein polynomials provide one shape parameter for constructing freeform curves and surfaces, Phillips q-Bernstein operator was applied well in this area, as Khan and Lobiyal [11] indicated before.
Leonid V. Kantorovich, in 1930, introduced a generalization of Bernstein operators as (1.2) MURAT BODUR -NES ˙IBE MANAV -FATMA TAS ¸DELEN K n operators are obtained from the classical Bernstein operators (1.1), by replacing the approximating function value f ( k n ) with integral of f in a neighborhood of k n [10].In 1968, D. D. Stancu [16] constructed positive linear operators S α,β n : Here f ∈ C[0, 1], α and β real parameters such that 0 ≤ α ≤ β and n ∈ N. Note that for α = β = 0 the operators turn to classical Bernstein operators [5].Some recent papers mentioning different types of generalizations of Benstein operators and its applications are cited in [11] and [13].
Constructing a new kind of Bézier curve or a new type of Bernstein operators by using different basis is currently of high interest to researchers.In 2010, Zhangxiang Ye and his colleagues [18] proposed their study not only constructing a new kind of basis functions of Bézier curve with single shape parameter, but also providing a practical algorithm of curve modeling.They discussed some important properties of the basis functions and the corresponding curves.Thus, they explained why a curve or a surface with shape parameters is a good choice.
In 2018, inspiring by a paper in [18], Cai et al. [6] introduced a new generalization of Bernstein operators as follows where λ ∈ [−1, 1] and bn,k for k = 0, 1, . . ., are defined by bn,0 (λ; It is easy to verify that n k=0 bn,k (λ; x) = 1 and these operators become the classical Bernstein operators for λ = 0.They obtained some approximation properties of λ-Bernstein operators and established an asymptotic formula.Also, they gave some numerical examples to show the rate of convergence of B n,λ (f ; x) to f (x).Very recently Acu et al. [1] presented a Kantorovich modification of λ-Bernstein operators.They considered the positive linear operators as (1.7) They proved a quantitative Voronovskaja type theorem by means of Ditzian-Totik modulus of smoothness.Also, a Grüss-Voronovskaja type theorem for λ-Kantorovich operators were given by them.They studied approximation properties of the Kantorovich variant of λ-Bernstein operators.Some articles can be given about related work [7,8,15].According to this progress, we defined Kantorovich-Stancu type λ-Bernstein operators as below where Bézier bases bn,k (λ; x) are defined in (1.6).Here, as we mentioned above α and β real parameters such that 0 ≤ α ≤ β and n ∈ N. Besides, it should be taken into consideration that (i) if we chose α = β = 0, the sequence of operators defined in (1.8) reduce to operators in (1.7), (ii) if we take λ = 0 in (1.The present work is organized as follows: We demonstrate moments of (1.8) operators can be obtained using the concept of moment generating function.A uniform of these constructed operators (1.8) is mentioned.After, we investigate order of approximation in the sense of local approximation with using a classical approach, the second modulus of continuity, Peetre's K-functional and Lipschitz type function.Then, a Voronovskaja type result is presented for given operators.Some graphical examples are also given in this paper in order to show the rate of convergence of (1.8) operators.

Preliminary results
In this part, we present the moments and the central moments of Kantorovich-Stancu type λ-Bernstein operators to use the proofs of the main theorems. (2.2)

APPROXIMATION PROPERTIES OF λ-BERNSTEIN-KANTOROVICH-STANCU OPERATORS
Besides, the consecutive limits hold

Convergence results
In this main chapter, we search the rate of convergence by means of the modulus of continuity, Peetre's K-functional and elements of Lipschitz class.Here, C[0, 1] is the class of real valued functions defined on [0, 1] which is uniformly continuous with the norm f = sup It can be clearly obtained these three conditions using Lemma 2.1.
Here, we will give some definitions to use in this section.
The positive constant M is independent of f and δ.
In the following theorem, the modulus of continuity is used to estimate the order of approximation to the function f .
P r o o f.Keeping in mind the following property of modulus of continuity We obtain Lastly, choosing δ = σ(α, β, n, λ)(x), so Hence, the proof is completed.
) dy, so, applying the operators at above equality, we get Taking into account the following property of modulus of continuity we can write By using (3.2) in (3.1), we get Using Cauchy-Schwarz inequality then we reach Choosing δ = σ(α, β, n, λ)(x), we find the desired inequality.Now, we interest in the rate of convergence by the known method K-functional.

APPROXIMATION PROPERTIES OF λ-BERNSTEIN-KANTOROVICH-STANCU OPERATORS
where M is a positive constant.
This implies that x ε α,β n,λ (x) − y g (y)dy In view of (3.3), we obtain Now, for f ∈ C[0, 1] and g ∈ W 2 [0, 1], using (3.3) and (3.4), we get Taking the infimum on the right side over all g ∈ W 2 [0, 1], we have Conclusively, when using the relation between K-functional and the second modulus of continuity as shown in Definition 3, we reach the desired result.
We investigate the rate of convergence with the help of the functions of Lipschitz class.For this, we have to give below definition primarily.
P r o o f.For the positive linear operators S α,β n,λ (f ; x) and f ∈ Lip M (γ), while and keeping in mind well-known Hölder's inequality, it can be written directly Hence, the proof is completed.

Voronovskaja type theorem
Now, we prove the Voronovskaja type theorem for the operators S α,β n,λ .P r o o f.The Taylor's expansion where h(.) ∈ C[0, 1] and lim t→x h(t; x) = 0. Taking into account the linearity of the operators S α,β n,λ and applying it to both sides of the above Taylor's expansion with a simple calculations, we obtain Therefore, by using Lemma 2.2, we get Hence, by the Cauchy-Schwarz inequality, we have n|S α,β n,λ (h(t, x)(t − x)

Graphical results
Finally, in this section , we provide some graphical examples that show the convergence of Kantorovich-Stancu type λ-Bernstein operators.With these given graphicals examples, it is possible to understand better how our operators converge to given functions.

Figure 4 .
Figure 4. Comparison between two the convergence of operators