Approximation by Bézier Variant of Baskakov-Durrmeyer-Type Hybrid Operators

Lahsen Aharouch, Khursheed J. Ansari , and M. Mursaleen 3,4 Department of Mathematics, College of Science, King Khalid University, 61413 Abha, Saudi Arabia Polydisciplinary Faculty of Ouarzazate, P.O. Box: 638, Ouarzazate, Morocco Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India


Introduction
To approximate continuous functions, many approximating operators have been introduced under certain conditions and with different parameters too. Many researchers have later generalized and modified these introduced operators and discussed various approximating properties of these operators. In 1957, Baskakov [1] introduced and studied such a class of positive linear operators, called Baskakov operators defined on the positive semiaxis. For f ∈ C½0,∞Þ, the sequence of Baskakov operators is given as for y ∈ ½0,∞Þ and n ∈ ℕ. Later on, many authors have been considering the Baskakov operators; for instance, Aral in [2] defines the parametric generalization of Baskakov operators as where P v n,k x ð Þ = x k−1 with n + k − 1 Among interesting studies realized in this context, we cite those based on the Baskakov-Kantorovitch-type operators in the generalized form (the original operator given by Kantorovich in [3]) defined as, for f ∈ L 1 ð½0, 1Þ (the class of Lebesgue integrable functions on ½0, 1), where χ n,k is the characteristic function of the interval ½k/n, k + 1/n. It is well known that Bézier curves are the mathematically defined curves successively used in computer-aided geometric design (CAGD), image processing, and curve fitting. The miscellaneous Bézier variant of operators is crucial subject matter in approximation theory. In 1983, Chang [4] pioneered the Bernstein-Bézier operators. Afterwards, several researchers established the Bézier variant of various operators (c.f. [5,6]). For more details on the approximation by Durrmeyer-type and Baskakov-Durrmeyer-type operators, one can refer to [7,8], respectively. For more about Bézier variant of operators, one can refer to [9,10].
We will be mainly interested to the Bézier variant operator type based on those of Baskakov-Durrmeyer defined as follows: where If we take θ = 1, then operator (5) reduces to the following operator studied by [11].
Let us briefly summarize the outline of the paper. Next section is devoted to the computation of some auxiliary results which we need to prove our theorems in coming sections. In Section 3, we will prove some approximations of functions using Ditzian-Totik modulus and then we will deal to functions lie in the Lipschitz spaces. We treat in Section 4 the rate of convergence in the context of suitable weighted spaces and functions having a derivative of bounded variation. Finally, in Section 5, we state and prove the quantitative Voronovskaja-type theorem.

Preliminary Results
Lemma 1. ξ v n,k ðxÞ satisfies the following important properties: (1) and (2) are evident, we prove only the assertion (3). If θ ≥ 1, it suffices to remark that by the mean value theorem, we have If θ < 1, we shall prove that Dividing this inequality by a θ , it is equivalent to prove that We have f ′ðrÞ = ðθ/ðr − 1ÞÞe θ ln ðr−1Þ − ðθ/rÞe θ ln ðrÞ ; then, and this is true as θ < 1.
We proved then f is increasing, so f ðrÞ > f ðsÞ for all r > s > 1, letting s to 1, and we deduce that f ðrÞ ≥ 0: ☐ Remark 2. The operators G v,θ n,ρ ð f ; xÞ have the integral representation where K v,θ n,k ðx, uÞ is the kernal defined by δðuÞ is the Dirac-delta function.
As an easy consequence of last lemma, we will prove the following result.
(1) On the one hand, we have On the other hand, (2) We have Using Lemma 1, it is easy to see that

Direct Approximation
Before we discuss the different approximations, we need some definitions. First, we recall the definition of the wellknown Ditizian-Totik modulus of smoothness w φ τ ð:, :Þ and Peetre's K-functional [12].
and the K-functional 3 Journal of Function Spaces where with AC loc is the set of all absolutely continuous function on every finite subinterval of ℝ + 0 .
Remark 9. w φ τ ð f , δÞ and K φ τ ðf , δÞ are equivalent, that is, there exists a constant C > 0 such that In the next definition, we cite Lipschitz-type functions: Definition 10 [13]. For a ≥ 0, b > 0 to be fixed, the class of two parametric Lipschitz-type functions is defined as where M is any positive constant and 0 < β ≤ 1: The space Lip 0,1 M ðβÞ is the space Lip * M ðβÞ given by Szász [14].
We now proceed with the approximation results.
Theorem 11. For f ∈ C B ðℝ + 0 Þ, we have where w φ τ is given by (20) and C is a constant free from the choice of n and x.
For the proof of this theorem, we use the following lemma proved in [15].
Proof. Let f ∈ Lip a,b M ðβÞ and x ∈ ð0+∞Þ, and we have

Journal of Function Spaces
Let us consider the case β = 1. By the Cauchy-Schwarz inequality and the fact G v,θ n,ρ ð1 ; xÞ = 1, we have immediately that This proves the result for β = 1.

Rate of Convergence in Weighted Spaces
In this section, we focus about the rate of convergence of operators (5) in the context of suitable weighted function spaces and functions having a derivative of bounded variation. We will use the following spaces: ð39Þ Introduce also These spaces are endowed with the norm The weighted modulus of continuity is defined as (see [16]) Theorem 14. Let f ∈ C * 2 ðℝ + 0 Þ. Then, for x ∈ ℝ + 0 , ρ, δ > 0, θ ≥ 1 and for large enough n, we have where C 1 , C 2 > 1 are constants independent of x and n: Proof. Let u, x ∈ ℝ + 0 , δ > 0. An immediate consequence of the definition of weighted modulus of continuity is Since G v,θ n,ρ ðf ; xÞ is linear and increasing, we have from (44) Cauchy-Schwarz inequality was applied in the last term, and it gives us Choosing δ = 1/ ffiffiffi n p , we get the required result in virtue of Remark 6. ☐

Rate of Convergence for Functions of Bounded Variation
Let DBVðℝ + 0 Þ be the space of functions on ℝ + 0 having a derivative of bounded variation on every finite subinterval of ℝ + 0 . Consider the space It is known that every function f in DBV 2 ðℝ + 0 Þ has a representation of the form where g is a function of bounded variation on each finite subinterval of ℝ + 0 .
Lemma 15. Let x ∈ ℝ + 0 , and let K v,θ n,ρ ðx, uÞ be the kernel defined by (13). Then, for C 1 > 1 and for n large enough, we have Proof. Using Remark 6, we get Similarly, we can show the second part; hence, the proof is omitted. ☐ Theorem 16. Let f ∈ DBV 2 ðℝ + 0 Þ, and for every x ∈ ð0, ∞Þ, consider the function f x ′ defined by Let us denote by ∨ d c f x ′ the total variation of f x ′ on ½c, d ⊂ ℝ + 0 . Then, for every x ∈ ð0, ∞Þ and large n, Proof. For any f ∈ DBV 2 ðℝ + 0 Þ, from the definition of f x ′ðuÞ, we can write where By the fact that G v,θ n,ρ ð1 ; xÞ = 1, we have From (52), we obtain Journal of Function Spaces From the definition of δ x ðvÞ, it is clear that The first integral on the right hand side of (55) can be estimated as follows: Applying the Cauchy-Schwarz inequality and Remark 2, we have, for n large enough, Similarly, it is easy to find Write the last term of (55) as Using the definition of ξ v,θ n,ρ ð:;:Þ given in Lemma 15 and integrating by parts, we can write