Approximation of functions by a class of Durrmeyer–Stancu type operators which includes Euler’s beta function

*Correspondence: mohiuddine@gmail.com 3Department of General Required Courses, Mathematics, Faculty of Applied Studies, King Abdulaziz University, Jeddah 21589, Saudi Arabia 1Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Full list of author information is available at the end of the article Abstract


Introduction
Let L B [0, 1] denote the space of bounded Lebesgue integrable functions on [0, 1] and N the set of natural numbers. We use the symbol m (m ∈ N) to denote the space of polynomials of degree at most m. By taking Bernstein polynomials into account, Chen [14] and Goodman and Sharma [21] independently introduced the operators U m (we can also call them genuine Bernstein-Durrmeyer operators) acting from L B [0,1]  The above operators are limits of the Bernstein-Durrmeyer operators with Jacobi weights, M c,d m for c, d > -1, which was studied by Păltănea [40], that is, Păltănea [41] presented a generalization of the operators U m with the help of ρ > 0, namely genuine ρ-Bernstein-Durrmeyer operators, and denoted them by U ρ m . For any f ∈ C[0, 1], in the same paper, he showed that the classical Bernstein operators are the limits of the operators U ρ m and also obtained a Voronovskaja-type result. Gonska and Păltănea [17] proved that the operators U ρ m preserve convexity of all orders and also obtained the degree of simultaneous approximation.
It is well known that Bernstein polynomials are one of the most widely-investigated polynomials in the theory of approximation, and so, to obtain another generalization of classical Bernstein operators, Cai et al. [13] considered the Bézier bases with shape parameter λ in [-1, 1] and introduced λ-Bernstein operators. Later, Kantorovich, Schurer, and Stancu variants of λ-Bernstein operators were discussed by Cai [11], Özger [36][37][38], and Srivastava et al. [43]. By taking λ-Bernstein polynomials into account, in a very recent past, Acu et al. [4] defined a new family of modified U ρ m operators and denoted the new operators by U ρ m,λ . Chen et al. [15] recently presented a generalization of classical Bernstein operators with the help of any fixed α in R, which they called α-Bernstein operators (linear and positive for α ∈ [0, 1]), and discussed the rate of convergence, Voronovskaja-type formula, and shape preserving properties of these positive linear operators. Mohiuddine et al. [26] constructed the Kantorovich variant of α-Bernstein operators. The bivariate version of α-Bernstein-Durrmeyer operators was constructed and studied by Kajla and Miclăuş [23] (also see [25] for recent work), in which they also discussed GBS operator (or generalized boolean sum operators) of α-Bernstein-Durrmeyer, while the two interesting forms of α-Baskakov-Durrmeyer were introduced by Kajla et al. [24] and Mohiuddine et al. [31]. For the classical Bernstein-Durrmeyer operators, we refer the interested reader to [16]. We also refer to [2, 3, 7, 8, 10, 12, 18, 19, 22, 27-30, 32-35, 39, 42, 45, 46] for some recent work on various Bernstein, Durrmeyer, and genuine type operators as well as statistical approximation.

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Generalized U ρ m operators and approximation properties
For m ∈ N and ρ > 0, the functional (see [41]) and Euler's beta function in the last equality is defined by Assume that θ and β are two real parameters satisfying 0 ≤ θ ≤ β. In view of α-Bernstein operators, for m ∈ N, α ∈ R is fixed, and given a function g ∈ C[0, 1], we define the operators U β,θ,ρ m,α (or genuine (α, ρ)-Durrmeyer-Stancu operators) by . Consequently, we can rewrite our operators U β,θ,ρ m,α as follows: For the choice of θ = 0 and β = 0, the operators defined by (2.3) reduce to the operators U ρ m,α (g; y) which were studied in [6]. In addition, if ρ = 1, then we get the genuine α-Bernstein-Durrmeyer operators U m,α defined in [1]. If we take ρ = 1, α = 1, θ = 0, and β = 0, then we obtain genuine Bernstein-Durrmeyer operators. Throughout the paper, we assume that α ∈ [0, 1] for which our new operators U β,θ,ρ m,α are linear and positive. For interested readers who want to see the details of Stancu operators, we refer to [44].
The moments of our newly constructed operators U β,θ,ρ m,α are given in the following lemma.
Using the properties of Euler beta function, we have

Voronovskaja-type theorems
We obtain some Voronovskaja-type theorems including a Grüss-Voronovskaja-type theorem and a quantitative Voronovskaja-type theorem for U β,θ,ρ m,α . We first obtain a quantitative Voronovskaja-type theorem for our operators U β,θ,ρ m,α using the Ditzian-Totik modulus of smoothness. To do this, we need the following definitions.
Proof The following equality 1]. This equality implies If we apply the operators U β,θ,ρ m,α to each side of (3.1), we get Let us estimate the right-hand side of (3.2) as follows: hold for sufficiently large m. Using the Cauchy-Schwarz inequality, one obtains U β,θ,ρ m,α (g; y)g(y) -U β,θ,ρ m,α (ty); y g (y) - Considering inf g∈W φ [0,1] on the right-hand side of the last inequality, we deduce the desired result.
Thus we immediately obtain the desired result by applying limit to (3.5) and by considering Corollary 1.

Numerical analysis
With the help of MATHEMATICA, we numerically examine our theoretical results with a view of convergence and error of approximation of our newly constructed operators (2.3). We first choose the parameters β, θ , ρ, α as β = 0.2, θ = 0.1, ρ = 1.5, α = 0.9 and the function g(y) = cos(2πy).
In Fig. 1, we examine the convergence of (2.3) for different m values, and in Fig. 2, we compare the convergence of our operators with U ρ m,α . We also study the approximation properties of (2.3) by considering the following function: g(y) = y|y -y 3 | y 3 + 1 2 y ∈ [0, 1] .