- Research
- Open access
- Published:
Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators
Advances in Difference Equations volume 2020, Article number: 676 (2020)
Abstract
We construct the bivariate form of Bernstein–Schurer operators based on parameter α. We establish the Voronovskaja-type theorem and give an estimate of the order of approximation with the help of Peetre’s K-functional of our newly defined operators. Moreover, we define the associated generalized Boolean sum (shortly, GBS) operators and estimate the rate of convergence by means of mixed modulus of smoothness. Finally, the order of approximation for Bögel differentiable function of our GBS operators is presented.
1 Introduction
Schurer [41] presented the modification of the classical Bernstein operators with the help of nonnegative parameter and nowadays called Bernstein–Schurer operators, which are linear and positive. Suppose that \(\mathbb{Z}_{0}^{+}\) and \(C[a,b]\) are used to denote the space of nonnegative integers and continuous functions on \([a,b]\), respectively. Let us take \(\eta \in \mathbb{Z}_{0}^{+}\). The well-known Bernstein–Schurer operators
are defined as
for any \(g\in [0,1+\eta ]\), \(j\in \mathbb{N}\), and \(y\in [0,1]\), where
When \(\eta =0\) in (1.1), we obtain
In this case, the operators \(M_{j}(g;y)\) and \(M_{j,0,k}(y)\), respectively, are called Bernstein operators and polynomials [11].
Most recently, the generalization of Bernstein operators was demonstrated by the authors Chen et al. [17] by taking the parameter \(\alpha \in \mathbb{R}\). However, they showed that their operators are positive and linear for the choice of \(0\leq \alpha \leq 1\), so they considered this assumption in their work and then studied several approximation properties for their α-Bernstein operators. Thereafter, many researchers, keeping the idea of this meaningful parameter α into account, constructed several operators. For example, Mohiuddine et al. [26] introduced the family of α-Bernstein–Kantorovich operators and associated bivariate form and demonstrated the results regarding the rate of convergence via Peetre’s K-functional together with modulus of continuity. In addition to this, the Stancu type α-Bernstein–Kantorovich, α-Baskakov and their Kantorovich form, and α-Baskakov–Durrmeyer operators were analyzed by Mohiuddine and Özger [32], Aral et al. [6, 20], and Mohiuddine et al. [31], respectively, and for other blending type operators, see [23, 27, 36]. Some other modifications of Bernstein operators have been studied in [2, 15, 16, 25, 33, 34, 37, 42]. Furthermore, Acar and Kajla [3] gave the bivariate α-Bernstein operators and associated generalized Boolean sum operators and then studied the degree of approximation of their operators. For more details on approximation by related operators and statistical approximation, we refer to [1, 4, 10, 14, 21, 28–30, 35, 40, 44, 45].
Motivated by the operators defined in (1.1) and α-Bernstein operators, very recently, Ozger et al. [38] defined the α-Bernstein–Schurer operators \(M_{j,\eta }^{\alpha }:C[0,1+\eta ]\longrightarrow C[0,1]\) by
and
for any \(g\in C[0,1+\eta ]\), \(y\in [0,1]\), \(j\in \mathbb{N}\), and \(0\leq \alpha \leq 1\), where
and
for \(j\geq 2\). Note that \(M_{j,\eta }^{\alpha } (t-y;y )=\eta y/j\). When \(\eta =0\), the operators \(M_{j,\eta }^{\alpha } (g;y )\) coincide with α-Bernstein operators. In addition to \(\eta =0\), take \(\alpha =1\), then \(M_{j,\eta }^{\alpha } (g;y )\) reduces to \(M_{j}(g;y)\). When only \(\alpha =1\), the operators \(M_{j,\eta }^{\alpha } (g;y )\) reduce to the operators \(M_{j,\eta }(g;y)\). In the same paper, authors investigated global approximation, local approximation, and Voronovskaja-type approximation results of the operators \(M_{j,\eta }^{\alpha } (g;y )\). They also established shape preserving properties such as monotonicity and convexity.
2 Bivariate generalized Bernstein–Schurer operators
Here, we construct the bivariate form of α-Bernstein–Schurer operators and demonstrate their basic properties.
Throughout the manuscript, we suppose that \(C(I^{2})\) is the space of continuous function on \(I^{2}(=I\times I)=[0,1+\eta ]\times [0,1+\eta ]\), where η in \(\mathbb{Z}_{0}^{+}\). For any \(h\in C(I^{2})\), \((y_{1},y_{2})\in [0,1]\times [0,1]\), \(s_{1},s_{2}\in \mathbb{N}\), and \(\alpha _{1},\alpha _{2}\) in \([0,1]\), we define
where the polynomials \(M_{s_{1}+\eta ,s_{2}+\eta ,\ell _{1},\ell _{2}}^{(\alpha _{1}, \alpha _{2})} (y_{1},y_{2} )=M_{s_{1}+\eta ,\ell _{1}}^{( \alpha _{1})} (y_{1} )M_{s_{2}+\eta ,\ell _{2}}^{(\alpha _{2})} (y_{2} )\) are considered by
Lemma 2.1
Suppose that \(e_{jk}(y_{1},y_{2})=y_{1}^{j}y_{2}^{k}\) for \((j,k)=\mathbb{N}_{0}\times \mathbb{N}_{0}\) \((\mathbb{N}_{0}=\mathbb{N}\cup \{0\})\) with \(j+k\leq 4\). Then
Proof
We shall use Lemma 3 of [38] to prove Lemma 2.1. Clearly, from (2.1), we obtain
Next, we write
Similarly, we get
Further,
and similarly, we obtain
Similarly, we obtain the last two moments. □
Corollary 2.2
The following identities hold:
3 Order of convergence and Voronovskaja-type results
In this section, we obtain the Voronovskaja-type result and the order of convergence with the help of Peetre’s K-functional for our operators \(M_{s_{1},s_{2},\eta }^{\alpha _{1},\alpha _{2}} (h;y_{1},y_{2} )\). For \(g\in C (I^{2} )\), the norm of the bivariate function g is considered by
Theorem 3.1
For any \(h\in C(I^{2})\), one has
where \(I_{1}^{2}=[0,1]\times [0,1]\).
Proof
We see that
as \(s_{1},s_{2}\to \infty \). Thus (3.1) holds by Volkov’s theorem [43]. □
We will use \(C^{2}(I^{2})\) to denote the space of all functions \(h\in C(I^{2})\) such that \(\frac{\partial ^{j}h}{\partial y_{1}^{j}}, \frac{\partial ^{j}h}{\partial y_{2}^{j}}, \frac{\partial ^{2}h}{\partial y_{1}\,\partial y_{2}}\in C(I^{2})\) \((j=1,2)\) and equipped with the norm
Theorem 3.2
Suppose that \(h\in C^{2}(I^{2})\). Then
uniformly on \(I_{1}^{2}\).
Proof
Suppose that \(h\in C^{2}(I^{2})\). Then Taylor’s theorem gives
where \(\rho (t_{1},t_{2},y_{1},y_{2})\in C(I^{2})\) and
Since \(M_{s,s,\eta }^{\alpha _{1},\alpha _{2}}\) is linear so, by operating on (3.2), we obtain
With a view of Corollary 2.2, we find that
and also
It follows from (3.3)–(3.6) that
By the Cauchy–Schwarz inequality, one gets
Since \(\rho (t_{1},t_{2},y_{1},y_{2})\in C(I^{2})\) and \(\lim_{ (t_{1},t_{2} )\to (y_{1},y_{2} )} \rho (t_{1},t_{2},y_{1},y_{2})=0\), we have
uniformly on \(I_{1}^{2}\) by Theorem 3.1. Corollary 2.2 gives
and
Employing the last three relations in Eq. (3.8) and then using Eq. (3.7), we obtain
uniformly on \(I_{1}^{2}\). □
For \(h\in C(I^{2})\) and \(\delta >0\), Peetre’s K-functional is given by
and the modulus of continuity of h is
By Theorem 9 (see [18]), there is a constant \(C>0\) such that
In the above relation, \(\omega _{2}(h;\sqrt{\delta })\) is the second-order modulus of continuity of \(h\in C(I^{2})\) (for details, see [5]).
Theorem 3.3
For any \(h\in C(I^{2})\), one has
where
and
Proof
Assume that \(h_{1}\in C^{2}(I^{2})\). Then, by Taylor’s theorem, we have
We are now defining the auxiliary operators by
Simple calculation together with Corollary 2.2 gives that
By operating \({\mathcal{M}}_{s_{1},s_{2},\eta }^{\alpha _{1},\alpha _{2}}\) in (3.10) and using the last relation, we obtain
which yields
We can see that
which yields \(\vert {\mathcal{M}}_{s_{1},s_{2},\eta }^{\alpha _{1},\alpha _{2}} (h;y_{1},y_{2} ) \vert \leq 3\|h\|\) for any \(h\in C(I^{2})\). We therefore write
Now, taking \(\inf_{h_{1}\in C^{2}(I^{2})}\), we get
which completes the proof. □
The following corollary follows from Theorem 3.3 and inequality (3.9).
Corollary 3.4
Let \(h\in C(I^{2})\). Then
4 GBS operators of bivariate generalized Bernstein–Schurer type
For any compact real intervals X and Y, the function \(h:Y_{1}\times Y_{2}\to \mathbb{R}\) is B-bounded (or Bögel bounded) on \(Y_{1}\times Y_{2}\) if there exists \(H>0\) such that
where \(\Delta _{(y_{1},y_{2})}h [u_{1},u_{2};y_{1},y_{2} ]\) is the mixed difference of h defined by
A function h is said to be B-continuous (or Bögel continuous) at a point \((u_{1},u_{2})\) if
for any \((u_{1},u_{2} ), (y_{1},y_{2} )\in Y_{1} \times Y_{2}\) (see [13]).
Given a function \(h\in C(I^{2})\), for any \(s_{1},s_{2}\in \mathbb{N}\), \(\eta \in \mathbb{Z}_{0}^{+}\), and \(\alpha _{1},\alpha _{2}\in [0,1]\), we define the generalized Boolean sum (or GBS) operators of the bivariate form of generalized Bernstein–Schurer operators (2.1) by
or, equivalently, we write
Note that \(\Phi _{s_{1},s_{2},\eta }^{\alpha _{1},\alpha _{2}} (h;y_{1},y_{2} )\) is well defined on \(C_{B}(I^{2})\) (the space of all B-continuous functions on \(I^{2}\)) into \(C(I^{2})\) and \(h\in C_{B}(I^{2})\) as well as linear and positive.
Recall that the mixed modulus of smoothness of \(h\in C_{B}(I^{2})\) is given by
for any \(\delta _{1},\delta _{2}>0\) (see [7, 9]).
A function h is B-differentiable (or Bögel differentiable) at the point \((u_{1},u_{2})\in Y_{1}\times Y_{2}\) if
exists. The limit is called B-differentiable of h at \((u_{1},u_{2})\) and denoted by \(D_{y_{1}y_{2}}h (u_{1},u_{2} )=D_{B} (h;u_{1},u_{2} )\). By \(D_{b}(Y_{1}\times Y_{2})\), we denote the set of all B-differentiable functions.
For more details and related results, we refer to [8, 12, 19, 22, 24, 39].
The following theorem gives an estimate of the rate of convergence of \(\Phi _{s_{1},s_{2},\eta }^{\alpha _{1},\alpha _{2}}\) to \(h\in C_{B}(I^{2})\).
Theorem 4.1
For any \(h\in C_{B}(I^{2})\), the inequality
holds, where
Proof
It follows from (4.3) and
that
for all \((y_{1},y_{2}),(t_{1},t_{2})\in I^{2}\) and for any \(\delta _{1},\delta _{2}>0\). Rewrite (4.1) as
Operating \(M_{s_{1},s_{2},\eta }^{\alpha _{1},\alpha _{2}}\) and using the definition of \(\Phi _{s_{1},s_{2},\eta }^{\alpha _{1},\alpha _{2}}\), we obtain
which yields
Employing (4.4) and then using the Cauchy–Schwarz inequality, we get
Using Corollary 2.2, we write
and
We therefore obtain
which gives the assertion of Theorem 4.1 by choosing \(\delta _{1}=\sqrt{\beta _{s_{1},\eta }}\) and \(\delta _{2}=\sqrt{\beta _{s_{2},\eta }}\). □
Finally, we study the order of approximation for B-differentiable functions of our operators \(\Phi _{s_{1},s_{2},\eta }^{\alpha _{1},\alpha _{2}}\).
Theorem 4.2
Suppose that \(h\in D_{b}(I^{2})\) and \(D_{B}h\) in \(B(I^{2})\) (the space of all bounded functions on \(I^{2}\)). Then
where \(N>0\) is a constant.
Proof
Let \(h\in D_{b}(I^{2})\). Then, from (see [13], p. 62), we write
for \(y_{1}<\alpha <t_{1}\), \(y_{2}<\gamma <t_{2}\). Thus, we fairly have
Since \(D_{B}h\in B(I^{2})\), we have \(|D_{B}h(y_{1},y_{2})|\leq \|D_{B}h\|_{\infty }\). In view of the last two equalities, we obtain
Also, we have
We thus have from (4.5) and (4.6) together with the Cauchy–Schwarz inequality that
which gives
By straightforward calculation (from Corollary 2.2), we obtain
for some constant \(N_{1},N_{2}>0\). Also
for \((t_{1}-y_{1} ), (t_{2}-y_{2} )\in I^{2}\) and \(m,n=1,2\). From the above and by choosing \(\delta _{1}=\sqrt{\frac{1}{s_{1}}}\) and \(\delta _{2}=\sqrt{\frac{1}{s_{2}}}\), we have
which yields
where
which completes the proof. □
Availability of data and materials
Not applicable.
References
Acar, T., Aral, A., Mohiuddine, S.A.: On Kantorovich modification of \((p,q)\)-Baskakov operators. J. Inequal. Appl. 2016, 98 (2016)
Acar, T., Aral, A., Mohiuddine, S.A.: Approximation by bivariate \((p,q)\)-Bernstein–Kantorovich operators. Iran. J. Sci. Technol. Trans. A, Sci. 42, 655–662 (2018)
Acar, T., Kajla, A.: Degree of approximation for bivariate generalized Bernstein type operators. Results Math. 73, 79 (2018)
Acar, T., Mohiuddine, S.A., Mursaleen, M.: Approximation by \((p,q)\)-Baskakov–Durrmeyer–Stancu operators. Complex Anal. Oper. Theory 12, 1453–1468 (2018)
Anastassiou, G.A., Gal, S.G.: Approximation Theory: Moduli of Continuity and Global Smoothness Preservation. Birkhäuser, Boston (2000)
Aral, A., Erbay, H.: Parametric generalization of Baskakov operators. Math. Commun. 24, 119–131 (2019)
Badea, C., Badea, I., Cottin, C., Gonska, H.H.: Notes on the degree of approximation of B-continuous and B-differentiable functions. Approx. Theory Appl. 4, 95–108 (1988)
Badea, C., Badea, I., Gonska, H.H.: A test function theorem and approximation by pseudo polynomials. Bull. Aust. Math. Soc. 34, 53–64 (1986)
Badea, C., Cottin, C.: Korovkin-type theorems for generalized boolean sum operators, approximation theory (Kecskemét, 1900). In: Colloq. Math. Soc. János Bolyai, vol. 58, pp. 51–68. North-Holland, Amsterdam (1990)
Belen, C., Mohiuddine, S.A.: Generalized weighted statistical convergence and application. Appl. Math. Comput. 219, 9821–9826 (2013)
Bernstein, S.N.: [Fr]Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun. Kharkov Math. Soc. 13, 1–2 (1912/1913)
Bögel, K.: Mehrdimensionale differentiation von funktionen mehrerer reeller Veränderlichen. J. Reine Angew. Math. 170, 197–217 (1934)
Bögel, K.: Über die mehrdimensionale differentiation. Jahresber. Dtsch. Math.-Ver. 65, 45–71 (1962)
Braha, N.L., Srivastava, H.M., Mohiuddine, S.A.: A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean. Appl. Math. Comput. 228, 162–169 (2014)
Cai, Q.-B.: The Bézier variant of Kantorovich type λ-Bernstein operators. J. Inequal. Appl. 2018, Article ID 90 (2018)
Cai, Q.-B., Lian, B.-Y., Zhou, G.: Approximation properties of λ-Bernstein operators. J. Inequal. Appl. 2018, Article ID 61 (2018)
Chen, X., Tan, J., Liu, Z., Xie, J.: Approximation of functions by a new family of generalized Bernstein operators. J. Math. Anal. Appl. 450, 244–261 (2017)
Devore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)
Dobrescu, E., Matei, I.: The approximation by Bernstein type polynomials of bidimensional continuous functions. An. Univ. Timişoara Ser. Şti. Mat.-Fiz. 4, 85–90 (1966)
İlarslan, H.G.I., Erbay, H., Aral, A.: Kantorovich-type generalization of parametric Baskakov operators. Math. Methods Appl. Sci. 42, 6580–6587 (2019)
Kadak, U., Mohiuddine, S.A.: Generalized statistically almost convergence based on the difference operator which includes the \((p,q)\)-gamma function and related approximation theorems. Results Math. 73, 9 (2018)
Kajla, A., Miclăuş, D.: Blending type approximation by GBS operators of generalized Bernstein–Durrmeyer type. Results Math. 73(1), Article ID 1 (2018)
Kajla, A., Mohiuddine, S.A., Alotaibi, A., Goyal, M., Singh, K.K.: Approximation by ϑ-Baskakov–Durrmeyer-type hybrid operators. Iran. J. Sci. Technol. Trans. A, Sci. 44, 1111–1118 (2020)
Miclăuş, D.: On the GBS Bernstein–Stancu’s type operators. Creative Math. Inform. 22, 73–80 (2013)
Mohiuddine, S.A., Acar, T., Alghamdi, M.A.: Genuine modified Bernstein–Durrmeyer operators. J. Inequal. Appl. 2018, 104 (2018)
Mohiuddine, S.A., Acar, T., Alotaibi, A.: Construction of a new family of Bernstein–Kantorovich operators. Math. Methods Appl. Sci. 40, 7749–7759 (2017)
Mohiuddine, S.A., Ahmad, N., Özger, F., Alotaibi, A., Hazarika, B.: Approximation by the parametric generalization of Baskakov–Kantorovich operators linking with Stancu operators. Iran. J. Sci. Technol. Trans. A, Sci. (2020). https://doi.org/10.1007/s40995-020-01024-w
Mohiuddine, S.A., Alamri, B.A.S.: Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 1955–1973 (2019)
Mohiuddine, S.A., Asiri, A., Hazarika, B.: Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems. Int. J. Gen. Syst. 48(5), 492–506 (2019)
Mohiuddine, S.A., Hazarika, B., Alghamdi, M.A.: Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems. Filomat 33(14), 4549–4560 (2019)
Mohiuddine, S.A., Kajla, A., Mursaleen, M., Alghamdi, M.A.: Blending type approximation by τ-Baskakov–Durrmeyer type hybrid operators. Adv. Differ. Equ. 2020, 467 (2020)
Mohiuddine, S.A., Özger, F.: Approximation of functions by Stancu variant of Bernstein–Kantorovich operators based on shape parameter α. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 70 (2020)
Mursaleen, M., Ansari, K.J., Khan, A.: Approximation by a Kantorovich type q-Bernstein–Stancu operators. Complex Anal. Oper. Theory 11(1), 85–107 (2017)
Mursaleen, M., Khan, F., Khan, A.: Approximation properties for modified q-Bernstein–Kantorovich operators. Numer. Funct. Anal. Optim. 36(9), 1178–1197 (2015)
Nasiruzzaman, M.: Approximation properties by Szász–Mirakjan operators to bivariate functions via Dunkl analogue. Iran. J. Sci. Technol. Trans. A, Sci. (2020). https://doi.org/10.1007/s40995-020-01018-8
Nasiruzzaman, M., Rao, N., Wazir, S., Kumar, R.: Approximation on parametric extension of Baskakov–Durrmeyer operators on weighted spaces. J. Inequal. Appl. 2019, 103 (2019)
Özger, F.: Weighted statistical approximation properties of univariate and bivariate λ-Kantorovich operators. Filomat 33(11), 3473–3486 (2019)
Özger, F., Srivastava, H.M., Mohiuddine, S.A.: Approximation of functions by a new class of generalized Bernstein–Schurer operators. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 173 (2020)
Pop, O.T.: Approximation of B-differentiable functions by GBS operators. An. Univ. Oradea, Fasc. Mat. 14, 15–31 (2007)
Rao, N., Nasiruzamman, M.: A generalized Dunkl type modifications of Phillips operators. J. Inequal. Appl. 2018, 323 (2018)
Schurer, F.: Linear positive operators in approximation theory. Math. Inst. Techn. Univ, Delft Report (1962)
Srivastava, H.M., Özger, F., Mohiuddine, S.A.: Construction of Stancu-type Bernstein operators based on Bézier bases with shape parameter λ. Symmetry 11(3), 316 (2019)
Volkov, V.I.: On the convergence of sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk SSSR 115, 17–19 (1957)
Wafi, A., Rao, N.: Szász-gamma operators based on Dunkl analogue. Iran. J. Sci. Technol. Trans. A, Sci. 43(1), 213–223 (2019)
Wafi, A., Rao, N.: Approximation properties of \((p,q)\)-variant of Stancu–Schurer operators. Bol. Soc. Parana. Mat. 37(4), 137–151 (2019)
Acknowledgements
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (D-025-130-1438). The author, therefore, gratefully acknowledges the DSR for technical and financial support.
Funding
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (D-025-130-1438).
Author information
Authors and Affiliations
Contributions
The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mohiuddine, S.A. Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators. Adv Differ Equ 2020, 676 (2020). https://doi.org/10.1186/s13662-020-03125-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-020-03125-7