Approximation by bivariate Chlodowsky type Szász–Durrmeyer operators and associated GBS operators on weighted spaces

In this article, we consider a bivariate Chlodowsky type Szász–Durrmeyer operators on weighted spaces. We obtain the rate of approximation in connection with the partial and complete modulus of continuity and also for the elements of the Lipschitz type class. Moreover, we examine the degree of convergence with regard to the weighted modulus of continuity and Peetre’s K-functional. Further, we construct the associated GBS type of these operators and estimate the degree of approximation using the mixed modulus of continuity and a class of the Lipschitz of Bögel type continuous functions. Finally, with the help of Maple software, we present the comparisons of the convergence of the bivariate Chlodowsky type Szász–Durrmeyer operators and associated GBS type operators to certain functions with some graphs and error estimation tables.


Introduction
The approximation of the continuous functions via the sequences of linear positive operators, which have many applications in disciplines such as engineering and physics, besides mathematics, has been an important research topic since the last century. In [1], Bernstein proposed one of the elegant proof of the Weierstrass approximation theorem. A generalization of Bernstein operators on an unbounded set was introduced by Chlodowsky [2]. In 1930, an integral modification of the classical Bernstein operators was presented by Kantorovich [3]. In [4,5], Szász-Mirakjan considered the linear positive operators on [0, ∞), which are related to the Poisson distribution. In 1957, Baskakov [6] studied a sequence of positive linear operators for the convenient functions defined on the interval [0, ∞). To approximate the Lebesgue integrable functions, Durrmeyer [7] introduced and studied the integral modification of the Bernstein operators. In recent years, a lot of generalizations and modifications of above-mentioned operators over finite or infinite intervals have been studied by several authors. One of the prominent components of approximation theory, the Korovkin type theorems, on weighted spaces was introduced by Gadzhiev [8,9]. Ispir [10] presented a modification of the Baskakov operators for the interval [0, b m ] and derived some approximation results in terms of the Korovkin theorems. In [11], Ditzian studied a necessary and adequate condition on the degree of convergence of Szász-Mirakjan and Baskakov operators on weighted spaces. In 2005, Ibikli and Karsli [12] proposed the Bernstein-Chlodowsky operators in terms of Durrmeyer type operators and reached some approximation results of these operators. Mazhar and Totik [13] considered a modification of the integral type of the Szász-Mirakjan operators and proved direct estimates and saturation results of these operators. In [14], Mursaleen and Ansari defined the Chlodowsky version of the Szász operators by the Brenke type polynomials and established degree of convergence with a classical method, Peetre's K -functional and secondorder modulus of continuity. Dogru [15] investigated some properties of the continuous functions on [0, ∞) by the modified positive linear operators. In 2013, Izgi [16] introduced and studied the following composition of the Chlodowsky and Százs-Durrmeyer operators on weighted spaces is a positive and increasing sequence with the following assumption: He investigated the uniform convergence, rate of approximation on weighted spaces, and proved a Voronovskaya type asymptotic formula for the operators (1.1). Recently, the univariate or bivariate cases of several well-known linear positive operators have been studied in many papers [17][18][19][20][21][22][23][24].
The structure of this research is organized as follows: In Sect. 2, we propose the bivariate extension of operators (1.1). We introduce the uniform convergence of these operators and estimate the order of approximation in terms of the partial and complete modulus of continuity for the elements of the Lipschitz type class, weighted modulus of continuity, and Peetre's K -functional, respectively. In Sect. 3, we discuss the associated GBS type of these operators and investigate the order of convergence by the mixed modulus of smoothness and the Lipschitz class of the Bögel continuous functions. In the final section, we present some graphs and error estimation tables to compare the convergence of bivariate and associated GBS type operators to certain functions.

Construction of the operators
Let I α n β m := [0, α n ] × [0, β m ] and the space C(I α n β m ) be the set of all real-valued functions of bivariate continuous on I α n β m . The weighted function is given by ρ(z, y) = 1 + z 2 + y 2 , (z, y) ∈ I α n β m .
It is endowed with the norm μ ρ = sup (z,y)∈I αnβm μ(z,y) ρ(z,y) . Moreover, by B ρ (I α n β m ), we denote the real-valued continuous functions on I α n β m and verify |μ(z, y)| ≤ C μ ρ(z, y); here C μ is fixed and depends just on μ. We also denote by C ρ (I α n β m ) the subspace of every continuous function depending on B ρ (I α n β m ), and by C a ρ the subspace of every functions μ ∈ C ρ (I α n β m ), satisfying lim (z,y)→∞ μ(z,y) ρ(z,y) = a, where a is a constant depending on μ. In what follows, let e u,v (z, y) = z u y v , (z, y) ∈ I α n β m , (u, v) ∈ N 0 × N 0 with 0 ≤ u, v ≤ 4 be the bivariate test functions. Now, based on the method of parametric extensions (see: [25,26]), we define twodimensional Chlodowsky-Szász-Durrmeyer operators as follows: Proof Since the proofs of (ii), (iv), and (vi) can be obtained with similar calculations, we will only prove (i), (iii), and (v).

Lemma 2.4 As a consequence of Lemma 2.3, we have
Proof In view of Lemma 2.3, one can obtain Analogously, the proof of the inequalities (ii) and (iv) can be obtained by the same methods; thus we get the desired result.
In the next theorem, with the help of theorems related to the weighted approximation of functions of several variables proved by Gadzhiev et al. [27], we show the uniform convergence of the operators given by (2.1) on I α n β m . Hence, Proof Taking into account the following relations from Lemma 2.2: Hence, we get Thus, we have lim n,m→∞ R n,m e 2 1,0 + e 2 0,1z 2 + y 2 ρ = 0.
Since all conditions of the bivariate Korovkin type theorem are satisfied, we arrive lim n,m→∞ R n,m (μ)μ ρ = 0, for all μ ∈ C a ρ (I α n β m ). Consequently, the proof is complete.
, we give the complete modulus of continuity as follows: where (μ, γ n, γ m ) verify the subsequent properties: For the integers z and y, the partial modulus of continuity is given by Let the space C 2 (I cd ) denote the functions of μ such that . For μ ∈ C(I cd ), the norm on C 2 (I cd ) and Peetre's K -functional are defined as follows: and respectively, where ζ > 0. Also, the following inequality: denotes the second order of the modulus of continuity, and D > 0 is an absolute constant independent of μ, ζ and * ω 2 .

Theorem 2.7
Suppose that the operators R n,m (μ; z, y) are given by (2.1), and μ ∈ C(I cd ). Then the following relation verifies where γ n = γ n (z) and γ m = γ m (y) are given as in Theorem 2.6.
With the help of the Lipschitz class, we will also estimate the order of approximation of operators (2.1). Let μ ∈ C(I cd ), (z, y), (t, s) ∈ I cd and ϕ 1 , ϕ 2 ∈ (0, 1] the class of Lipschitz for the bivariate case is given by (2.4) Theorem 2.8 Suppose that μ ∈ Lip L (μ; ϕ 1 , ϕ 2 ). Then for all (z, y) ∈ I cd , we obtain where δ n (z) and δ m (y) are given as in Theorem 2.6.
Next, we will examine the degree of approximation of functions μ ∈ C a ρ on I α n β m . Analogously as in [18], for each μ ∈ C a ρ , we consider the weighted modulus of continuity as below: n,m (μ; γ 1 , γ 2 ) = sup s)ρ(z, y) .
Proof Firstly, we consider the following auxiliary operators: It follows by Lemma 2.2 that R n,m (tz; z, y) = 0 and R n,m (sy; z, y) = 0.
Hence, the required result is obtained.
Let us now give some definitions that we will use in this section. A function μ : U ×V → R, where U, V are compact intervals of R. For any (z, y), (t 0 , s 0 ) ∈ U × V , the mixed difference of μ is given as If a real-valued function μ satisfies the following relation, it is called a Bögel-continuous (B-continuous) at (t 0 , s 0 ) ∈ U × V . lim (t 0 ,s 0 )→(z,y) φ (t 0 ,s 0 ) μ(z, y) = 0.
If the following limit denoted by D B μ(z, y) exists and is finite, then a function μ is called a Bögel-differentiable (B-differentiable) at (t 0 , s 0 ) ∈ U × V .
Note that by C b (U × V ) and D b (U × V ), we denote the sets of each B-continuous and B-differentiable functions on U × V , respectively. Considering the definition of Bcontinuous, one gets C(U × V ) ⊂ C b (U × V ), see [39] for details.
A function μ : Further, by B b (U × V ), we denote the space of all B-bounded functions on U × V , and it is endowed with the norm μ B = sup (z,y),(t 0 ,s 0 )∈U×V |φ (t 0 ,s 0 ) μ(z, y)|.
Exactly, for any (z, y) ∈ I cd and μ ∈ C b (I cd ), the GBS type operators related to the R n,m operators are defined as: It is clear that the operators given by (3.4) are positive and linear. Now, with the help of the mixed modulus of continuity, we will estimate the degree of approximation of operators (3.4).
If Fig. 2 and Table 2 are analyzed in detail, it becomes obvious that the GBS type operators (3.4) are approximated much better than operators (2.1).