Abstract
In this article, we establish an extension of the bivariate generalization of the -Bernstein type operators involving parameter and extension of GBS (Generalized Boolean Sum) operators of bivariate -Bernstein type. For the first operators, we state the Volkov-type theorem and we obtain a Voronovskaja type and investigate the degree of approximation by means of the Lipschitz type space. For the GBS type operators, we establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate -Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MATLAB software.
1. Introduction
Let with , and In 2018, Chen et al. [1] proposed a new generalization of Bernstein operators based on a fixed real parameter as where the basis functions are defined as
The authors studied the established of some Korovkin type approximation properties and the degree of approximation by means of the modulus of continuity, Voronovskaja-type results, and shape-preserving properties for these operators.
This work took the attention of researchers from approximation theory for a short time. Since that time, lots of researchers have put forth many relevant studies on this issue, and numerous articles can be given interrelated with their work [2–5].
In [6] were introduced the bivariate extension of the operators (1) and studied the degree of approximation in terms of the second order Ditzian-Totik modulus of continuity for two variables. A Kantorovich variant of the -Bernstein operators (1) was introduced and studied in [7]. Many authors also considered the univariate and bivariate positive linear operators and studied their approximation behavior; we refer the reader to articles (cf. [8–17]) and references therein. Now, we give some basic definitions based on the -calculus [18], which are used in this paper. Let and be any real numbers.
The -number is defined as and for ,
The -number is defined as and for
For the integers such that , the -binomial is defined as
For an integer , the -factorial is defined as and .
Cai et al. [19] considered the generalized Bernstein type operators based on parameters analogue and for fixed real parameter as where the basis functions are defined as and Note that for these operators reduce to -Bernstein operators (1) and for (7) educes to Bernstein operators defined in [20].
Therefore, linear operators, in particular the limit -Bernstein operator, are of significant interest for applications.
The purpose of this article is to present an extension of the bivariate -Bernstein type operators involving parameters and obtain the degree of approximation by means of the Lipschitz type space for two variables. Moreover, we consider the associated Generalized Boolean Sum (GBS) operators and study their degree of approximation in terms of the mixed modulus of smoothness for bivariate functions.
2. Construction of the Bivariate ,-Bernstein Type Operators
For , let be the space of all continuous functions on .
For and the bivariate extension of the operator (7) is defined by where is a sequence in satisfying
Further, where and are defined similarly as in (8).
It is easy to see that is bounded. We denote
Lemma 1 (see [19]). For the operators we have (i)(ii)(iii)
Let , with . In order to obtain the main results, we need the following lemmas:
Lemma 2. For the operators we have (i)(ii)(iii)(iv)(v)
Corollary 3. Applying Lemma 2, we have (i)(ii)(iii)(iv)(v)(vi)(vii)(viii)In what follows, the Volkov-type approximation theorem is proved for
Theorem 4. Let Then, we have
Proof. Using Lemma 2, it is obvious and uniformly on The result follows using [[21], Thm 2.1].
In order to discuss the next results, let us recall the definitions of modulus of continuity and partial modulus of continuity
Definition 5 (see [22]). For and , the full modulus of continuity in the bi-variate case is defined as For each fixed the partial modulus of continuity of with respect to is defined and respectively.
Theorem 6. For , we have where .
Proof. Using the facts that Then, we use the following property of the complete modulus of continuity: we get Applying Cauchy Schwarz inequality, we have By choosing we obtain the desired result.
Theorem 7. Let , then the following inequality holds:
Proof. Similarly to previous theorems, using relations (19) we obtain
Applying Cauchy Schwarz inequality with and , we obtain the desired result.
Now, we want to give the quantitative result in terms of the Lipschitz class functionals. For any functions and , the function is said to be in Lipschitz class if a such that where is the Euclidean norm and is a positive real constant.
The following theorem yields us an estimate of error for functions in , by the operators .
Theorem 8. Let Then for sufficiently large and and for all , there holds the inequality where is a constant.
Proof. From hypothesis, we have where Applying Hölder’s inequality and Corollary 3, we obtain which leads to the required result on applying Corollary 3.
Let denote the space of continuous functions on whose first-order partial derivatives and are also continuous on .
Our next result yields us the rate of approximation for continuously differentiable functions on by the operators .
Theorem 9. Let . Then for sufficiently large and , we have where is some positive constant.
Proof. For be arbitrary, we may write Hence, applying the operator on both sides of the above equation, we obtain
By using sup-norm on we get
Hence, applying the Cauchy-Schwarz inequality and Corollary 3, we obtain from which the desired result is immediate.
The following result yields the degree of approximation of by in terms of the partial modul of continuity of the partial derivatives of .
Theorem 10. Let . Then for sufficiently large and , we have where are the partial modul of continuity of for and is some positive constant.
Proof. If we use the mean value theorem in the following form, we have where and . Applying the operator to both sides, we deduce that
Since and are continuous in , there exist positive constants and such that and , for all . Hence, applying the Cauchy-Schwarz inequality, we obtain
Choosing we get the required result.
3. Construction of GBS Operator of Generalized Bernstein Type
In the last two decades, the study of generalized Boolean sum (GBS) operators of certain linear positive operators has attracted very much attention in the approximation theory. In early 1937 with Bögel [23], a great number of studies are performed related to these operators. To make an analysis in multidimensional spaces, Bögel [23] introduced the concepts of continuity and differentiability in a Bögel space. There are still many authors working on this subject. Agrawal et al. [24] studied the degree of approximation for bivariate Lupa-Durrmeyer type operators based on Pólya distribution with associated GBS operators. Further, Gupta et al. [25] introduced some GBS operators of discrete and integral types and examined the properties of approximations in a Bögel space. Recently, Kajla and Miclacus [16] studied the rate of approximation of Bögel continuous and Bögel differentiable functions by the GBS operators of Bernstein-Durrmeyer type operators. In the last year, Kumar and Shivam [26] constructed the bivariate Kantorovich-type sampling operator, involved with GBS operators, as well as estimation of the rate of convergence of the sequences of these operators.
A function is called B-continuous at point if for any , (see [23]). The function is B-bounded on if there exists such that for every
Throughout this article, denotes all B-bounded functions on . The space of all B-continuous functions is denoted by .
Motivated by the above authors, we construct the GBS operator of , who is defined as follows: for all More precisely, the -analogue -Bernstein type GBS operator is defined as follows: where the operator is well-defined on the space into and .
4. Degree of Approximation by
For , the mixed modulus of smoothness of is defined by and for any Using (45), we have
The basic results of were studied by Badea et al. [27, 28] and are similar to the properties of the usual modulus of continuity for bivariate functions. We shall obtain the rate of approximation of the operators (44) to in terms of the mixed modulus of continuity for two variables. For this, we apply the Shisha-Mond theorem for -continuous functions defined by Gonska [29] and Badea and Cottin [28].
Theorem 11. For every , at each point and sufficiently large and , the operator (44) satisfy the following results where is a positive constant depending on parameters and .
Proof. Using (45) and applying the inequality (46), we have for every and for any Taking the definition of , we may write
Applying the operator on both sides of the above inequality
Now, from Lemma 2, with the help of Cauchy-Schwarz inequality and Remark 1 (in that order), we obtain
Now, setting , the required result is obtained.
For , the Lipschitz class of Bögel continuous functions is defined as where and are the Euclidean norm.
In the next result, we obtain the degree of approximation of the operators for functions in the Lipschitz-class of Bgel continuous functions.
Theorem 12. If , then for sufficiently large and , we have where is a positive constant.
Proof. Using the equation (50), Hölder’s inequality, Lemma 2, and Lemma 1, we get Thus, we get the desired result.
5. Numerical Results and Discussions
Example 13. Let us choose , and Denote the error function of approximation by operators. The convergence of the bivariate Bernstein operators: (yellow), then (red) and (magenta) to (blue) will be illustrated in Figure 1.
Example 14. For and 30, and the convergence of to is illustrated in Figure 2. Denote the error function of approximation by operators. This example explains the convergence of the operators that are going to the function if the values of are increasing.
Comparative results are given in Figure 3, Tables 1 and 2, for the errors of the approximation of and to the functions for , and Note that (see Tables 1 and 2 and Figure 3) the GBS-Bernstein operator approximation outperforms others.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.