Approximation degree of Durrmeyer–Bézier type operators

Recently, a mixed hybrid operator, generalizing the well-known Phillips operators and Baskakov–Szász type operators, was introduced. In this paper, we study Bézier variant of these new operators. We investigate the degree of approximation of these operators by means of the Lipschitz class function, the modulus of continuity, and a weighted space. We study a direct approximation theorem by means of the unified Ditzian–Totik modulus of smoothness. Furthermore, the rate of convergence for functions having derivatives of bounded variation is discussed.


Introduction
For a continuous function h on [0, 1], Bernstein [1] defined a linear positive operator in order to provide a very simple and elegant proof of the Weierstrass approximation theorem, namely B n (h; x) = n k=0 n k x k (1x) n-k h k n , x ∈ [0, 1].
In order to approximate continuous functions on [0, ∞), Szász [2] introduced the operator provided the infinite series on the right-hand side converges. Later on, for h ∈ C[0, ∞) and 0 ≤ β < 1, Jain [3] proposed a modification of the operators given in (1.1), namely It is observed that the Jain operator (1.2) includes the Szász operator (1.1) as a special case for β = 0. Recently, Gupta and Greubel [4] also proposed the Durrmeyer type modification of the operators given in (1.2) as They showed that these operators converge to h without any restriction on β. The moments for these operators were obtained by using Tricomi's hypergeometric functions and Stirling numbers of first kind, and some approximation properties of these operators were proved.
In the literature, many authors have discussed the approximation behavior of different summation-integral type operators (see [5,6]). For 0 ≤ β < 1 and c ≥ 0, Acu and Gupta [7] introduced mixed Durrmeyer type operators for x ∈ [0, ∞) as They determined the degree of approximation by means of the modulus of continuity and a weighted space. The authors also studied the approximation of functions having derivatives equivalent with a function of bounded variation. It is observed that the operator defined by (1.3) has two special cases: (1) If φ n,c (x) = e -nx and β = 0, then the Phillips operators are obtained [8].
The aim of this paper is to investigate the weighted approximation properties and a direct approximation result by means of the Ditzian-Totik modulus of smoothness ω φ τ (h; t), 0 ≤ τ ≤ 1, and the rate of convergence for functions having a derivative of bounded variation for the operators given by (1.4). Throughout this paper, C denotes a constant which may be different at each occurrence.

Preliminaries
In the sequel, the following auxiliary results are used to prove the main results of the paper.

Remark 1 It is observed that
Let C B [0, ∞) denote the space of all continuous and bounded functions on [0, ∞), where the norm is defined by

Lemma 3 For every h
Lemma 3 can easily be proved using (2.1).
Remark 2 We observe that in view of the inequality Hence, from (1.4), we get

Main results
For x ∈ (0, ∞), t ∈ [0, ∞), and 0 < r ≤ 1, as we can see in Özarslan and Duman [28], the Lipschitz type space is defined as In the following theorem, we obtain the rate of convergence of the operators P β,c n,α for functions in Lip * M (r).
In the following, we present some weighted approximation results. First, we recall some basic notations.
Also, let The next theorem provides us the degree of approximation of P β,c n,α in terms of the classical modulus of continuity for the functions in the weighted space C 2 [0, ∞).
Proof From [29], for x ∈ [0, b] and t ≥ 0, we obtain Applying Remark 2 and the Cauchy-Schwarz inequality, we get To determine the rate of convergence for functions in C 0 2 [0, ∞), Yüksel and Ispir [6] introduced the weighted modulus of continuity as In the following lemma, we state the properties of the weighted modulus of continuity (h; δ).
Hence, using (3.4), we get Applying Theorem 2, we can find n 2 ∈ N such that for all n greater than equal to n 2 . Combining (3.3)-(3.6), we obtain This proves the required result.
In the following theorem, we establish the rate of convergence of the operators P β,c n,α in terms of the weighted modulus of continuity .

Theorem 4
Let h ∈ C 0 2 [0, ∞). If β = β(n) → 0 as n → ∞ and lim n→∞ nβ(n) = l ∈ R, then, for sufficiently large n, we have where C is a positive constant independent of h and n.
Proof For x ∈ (0, ∞) and δ > 0, using the definition of weighted modulus of continuity and Lemma 4, we have Applying P β,c n,α (·; x) to both sides of the above inequality, we can write From Lemma 2, for sufficiently large n, it follows that nμ β,c n,2 (x) ≤ Cx(cx + 2) and n 2 μ β,c n,4 (x) ≤ Cx 2 (cx + 2) 2 , (3.8) where C is a positive constant. Now, applying the Cauchy-Schwarz inequality in the last term of (3.7), we obtain Combining estimates (3.7)-(3.9) and taking we reach the required result. Now our aim is to discuss the rate of convergence in terms of the unified Ditzian-Totik modulus of smoothness ω φ τ (h, t), 0 ≤ τ ≤ 1. First, we define the Ditzian-Totik modulus of smoothness and the Peetre K -functional. Let φ(x) = √ x(2 + cx) and h ∈ C B [0, ∞). The modulus ω φ τ (h, t), 0 ≤ τ ≤ 1, is defined as and the appropriate K -functional is given by where W τ is the subspace of the space of locally absolutely continuous functions g on [0, ∞), with φ τ g < ∞. By [30, Theorem 2.1.1], there exists a constant N > 0 such that (3.10) for sufficiently large n, where C is independent of h and n.
where j is a function of bounded variation on each finite subinterval of [0, ∞). For this purpose, we use the following auxiliary result.

Lemma 5
For fixed u ∈ (0, ∞) and sufficiently large n, we have

20)
where C is a positive constant.

where b a h(x) represents the total variation of h on [a, b], M is a constant, and h x is defined by
where Since P β,c n,α (1; x) = 1, using (1.4), for every x ∈ (0, ∞), we get From (3.22) and (3.23), we get Obviously, Next, using (1.4), we get (3.25) and substituting y = xx/ √ n and applying Lemma 5, we get Thus, For t ≥ 2x, we have t ≤ 2(tx) and x ≤ tx. Now, using (3.8), we obtain

Conclusion
The Bézier variant of a sequence of mixed hybrid operators has been introduced and the rate of convergence by means of the Lipschitz class and the modulus of continuity has been established. The weighted approximation properties and a direct approximation theorem have been obtained. The approximation of functions with derivatives of bounded variation has been studied.