Abstract
In this study, we construct a Stancu-type generalization of bivariate Bernstein–Kantorovich operators that reproduce exponential functions. Then, we investigate some approximation results for these operators. We use test functions to prove a Korovkin-type convergence theorem. Then, we show the rate of convergence by the modulus of continuity and give a Voronovskaya-type theorem. We give a covergence comparison about bivariate Bernstein–Kantorovich–Stancu operators and their exponential form.
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1 Introduction
The goal of approximation theory is to approximate a target function using straightforward, computable, and more useful functions. In 1912, Bernstein [1] defined the Bernstein operators for every function on the interval \([0, 1]\). Later, the various generalizations of Bernstein polynomials were investigated in [2–4].
In addition to classical Bernstein polynomials, there are many studies on two-dimensional Bernstein polynomials and generalizations, such as [5]. Different types of Bernstein–Kantorovich operators have been studied in [6–9]. In [10], for \(n\in \mathbb{N}\), \(f\in L_{1}([0,1]\times [0,1])\) Pop and Farcas constructed two variable Bernstein–Kantorovich-type operators \(K_{n}:L_{1}(S)\to C([0,1]\times [0,1])\). For any \((x,y)\in S\), these operators are defined as:
where \(k,j\geq 0\).
In 2020, the Stancu variant of Bernstein–Kantorovich operators based on the shape parameter α was introduced [11]. Also, the Stancu variant of well-known operators such as Bernstein, Baskakov, and Szász was introduced in [12–19]. The bivariate form of Bernstein operators has been also studied in the literature (see, for instance, [20–27] references therein). One of these extensions is bivariate Bernstein–Kantorovich–Stancu operators. These operators are defined [28] for the functions \(f\in C(A) \), \(A=[0,1]\times [0,1]\) as
and
where \(s,t\in [0,1] \) and \(m,n\in \mathbb{N}\), \(\alpha =(\alpha _{1},\alpha _{2})\), \(\beta =(\beta _{1},\beta _{2})\), \(0\leq \alpha _{1}\leq \beta _{1}\), \(0\leq \alpha _{2}\leq \beta _{2}\). In [22], Aral et al. gave the modification of exponential forms of Bernstein operators as follows
where
and
They defined the relation of their operators between the classical Bernstein operators as
Here, the exponential function is symbolized as \(\exp_{\alpha }(x)=e^{\alpha x}\), for a real parameter \(\alpha > 0\). The generalization of Bernstein operators given by Aral et al. [22] is a particular case of the modification introduced by Morigi and Neamtu in [29].
In 2019, Aral et al. [30] gave the Bernstein–Kantorovich operators that reproduce exponential functions for \(n \in \mathbb{N} \) and \(\alpha ,\beta , \mu > 0\) and \(x \in [0, 1]\). They considered the operator \(\widetilde{K}_{n} :C [0, 1]\to [0, 1] \) for the functions \(f\in C[0,1] \) as
This paper consists of 6 sections. In Sect. 2, we give the definition of generalized bivariate Bernstein–Kantorovich–Stancu operators and we obtain some auxiliary results. In Sect. 3, we mention the rate of convergence with the help of the modulus of continuity. In Sect. 4, we present Voronovskaya-type results. In Sect. 5, we illustrate numerical examples with graphics. In Sect. 6, we give the conclusions.
2 Preliminaries
In this article, we construct bivariate Bernstein–Kantorovich–Stancu operators that reproduce exponential functions.
Definition 2.1
Let \(S_{\mu ,\nu}=\{(x,y)\in \mathbb{R}^{2};x,y\geq 0,r_{\mu}+r_{\nu} \leq 1\}\subset S \) for each \(m,n\in \mathbb{N}\) and \(\mu ,\nu > 0\). We define Bernstein–Kantorovich–Stancu operators for the functions \(f \in C(S_{\mu ,\nu})\) as
where \(s,t\in [0,1]\) and \(m,n\in \mathbb{N}\), and \(\alpha =(\alpha _{1},\alpha _{2})\), \(\beta =(\beta _{1},\beta _{2})\), \(0\leq \alpha _{1}\leq \beta _{1}\), \(0\leq \alpha _{2}\leq \beta _{2}\). Here,
and so
\(\mu ,\nu >0\) are real parameters and \(\exp_{i,j}^{\mu ,\nu}\) represents the exponential function defined by \(\exp_{i,j}^{\mu ,\nu}(t,s):=e^{i\mu t+j\nu s}\) for \(0\leq i,j \leq 4\).
Lemma 2.1
Let \(m,n\) ∈ \(\mathbb{N}\) and \((x,y)\in S_{\mu ,\nu}\). The following equalities hold:
Proof
By taking \(f(t,s)=1\) in (4), we obtain
By taking \(f(t,s)=e^{\mu t}\) in (4), we achieve
By taking \(f(t,s)=e^{2\mu t }\) in (4), we obtain
By taking \(f(t,s)=e^{3\mu t }\) in (4), we have
By taking \(f(t,s)=e^{4\mu t }\) in (4), we obtain
By taking \(f(t,s)=e^{\mu t+\nu s }\) in (4), we obtain
Other results can be obtained in a similar way. □
Theorem 2.1
Let \(\alpha ,\beta \in (0,\infty )\). Then, we have
for \((i,j)\in \{ (0,0),(1,0),(0,1),(2,0),(0,2) \} \).
Proof
Hereby, by choosing the test functions \(\exp_{i,j}^{\mu ,\nu}(t,s):=e^{i\mu t+j\nu s}\) for
\((i,j)\in \{(0,0),(1,0),(0,1)\}\), we obtain that
By choosing \((i,j)=(2,0)\) and \((i,j)=(0,2)\), in (4), respectively, we obtain
□
Theorem 2.2
Let \(\mu ,\nu \in (0,\infty )\) and \(f \in C(S_{\mu ,\nu})\), then \(\widetilde{K}_{m,n}^{\alpha ,\beta ,\mu ,\nu}(f;x,y)\) converges to f uniformly.
Proof
Applying the Korovkin theorem, and from (8), (9), (10), and (11),
where \((i,j)\in \{(0,0),(1,0),(0,1),(2,0),(0,2)\}\), we obtain the desired result. □
Lemma 2.2
For any \((x,y)\in S_{\mu ,\nu}\), we obtain the limits of the central moments as follows:
3 Rate of convergence
The modulus of continuity \(\omega (f,\delta )\) for two-dimensional functions is given as follows:
Theorem 3.1
Let \(f\in C(S_{\mu ,\nu})\). The following inequality holds
where
Proof
From the definition of the modulus of continuity, we have
By using the Mean Value Theorem, we obtain
Here, if we choose
we have
□
4 Voronovskaya-type theorem
In this section, we mention a Voronovskaya-type theorem for the \(\widetilde{K}_{m,n}^{\alpha ,\beta ,\mu ,\nu}(f;x,y)\). Let the inverse of the exponential function for the first variable t be denoted by \(\log _{\mu}^{\nu}\) and the inverse of the exponential function for the second variable s be shown as \(\log _{\nu}^{\mu}\).
Theorem 4.1
Let \(f\in C(S_{\mu ,\nu})\). We have
uniformly in \((x,y)\in S_{\mu ,\nu}\).
Proof
From Taylor’s expansion for \((x,y)\in S_{\mu ,\nu}\), we have
where \(R(f,t,s;x,y)\to 0\) as \((t,s)\to (x,y) \).
By applying the operator \(\widetilde{K}_{m,n}^{\alpha ,\beta ,\mu ,\nu}(.;x,y)\) to both sides of (18), we write
Hence, we have the following derivatives:
and by substituting (20) into (19) and then by taking the limit we obtain
By using equalities (12)–(17), we obtain
When we apply the Cauchy–Schwarz inequality to (21), we obtain
Since \(R(t,s;x,y)\to 0\) as \((t,s)\to (x,y)\),
is verified uniformly in \(C(S_{\mu ,\nu})\). By using (16) and (17), we achieve the desired result. □
5 Graphical and numerical analysis
In this section, we give a graphical and numerical analysis of \(\widetilde{K}_{m,n}^{\alpha ,\beta ,\mu ,\nu}(f;x,y)\) operators that illustrate the modeling of the approximation for the function f.
Example 5.1
Let \(f(x,y)=\frac{\cos(x+1)\cos(y+1)}{e^{x+y+5}}\) for \(x,y\in [0.1,0.9]\). In Fig. 1, we show the graphs of \(\widetilde{K}_{m,n}^{\alpha ,\beta ,\mu ,\nu}(f;x,y)\) operators for fixed \(\alpha _{1} = \alpha _{2}=\beta _{1} = \beta _{2} =1\), the various values of \(\mu =\nu \in \{1,2,3\}\) and \(m=n \in \{70,80,90\}\).
We calculate the maximum errors of \(\|\widetilde{K}_{m,n}^{\alpha ,\beta ,\mu ,\nu}(f)-f \|\) for the function \(f(x,y)= \frac{\cos(x+1)\cos(y+1)}{e^{x+y+5}}\) by choosing \(x=y \in [0.1,0.9]\) and step size \(h=0.1\) in Table 1 for \(m=n\in \{70,80,90\}\).
Example 5.2
Let \(f(x,y)=e^{x+y}\). We give in Fig. 2 the graphs for \(\widetilde{K}_{70,70}^{5,10,0.9,0.9}(f;x,y)\), \(\widetilde{K}_{70,70}^{10,20,0.9,0.9}(f;x,y)\), and \(\widetilde{K}_{70,70}^{25,50,0.9,0.9}(f;x,y)\) for \(x=y \in [0.1,0.9]\).
We calculate the maximum errors of \(\|\widetilde{K}_{m,n}^{\alpha ,\beta ,\mu ,\nu}(f)-f \|\) for the function \(f(x,y)=e^{x+y}\) by choosing \(x=y \in [0.1,0.9]\), \(\mu =\nu =1\) and step size \(h=0.1\) in Table 2 for \(m=n\in \{70,80,90\}\).
In Table 3, by choosing \(f(x,y)=e^{x+y}\), we give the comparison of \(\widetilde{K}_{m,n}^{\alpha ,\beta}(f;x,y)\) and our new bivariate Bernstein–Kantorovich–Stancu operators \(\widetilde{K}_{m,n}^{\alpha ,\beta ,\mu ,\nu}(f;x,y)\).
6 Conclusion
In this work, we construct the exponential bivariate Bernstein–Kantorovich–Stancu operators. Then, we calculate the rate of convergence with the modulus of continuity of the functions defined on \(C(S_{\mu ,\nu})\). Also, we give the Voronovskaya-type theorem. Finally, the error tables of the exponential bivariate Berntein–Kantorovich operators are given for different values of \(m,n,\mu ,\nu ,\alpha \), and β.
Data availability
All data generated or analyzed during this study are included in this published article.
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Funding
L.T. Su is supported by the Natural Science Foundation of Fujian Province of China (Grant No. 2020J01783) and the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant No. 2022C001R). This work is supported by Science and Technology Program of Quanzhou No. 2021N180S.
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Lian-Ta Su: Construction of main problem and funding. Kadir Kanat: Construction of main problem, computer data analysis. Melek Sofyalioglu: computer data analysis, illustrations of figures. Merve Kisakol: made calculations, wrote the main manuscript text. All authors reviewed the manuscript.
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Su, LT., Kanat, K., Sofyalioğlu Aksoy, M. et al. Approximation by bivariate Bernstein–Kantorovich–Stancu operators that reproduce exponential functions. J Inequal Appl 2024, 6 (2024). https://doi.org/10.1186/s13660-024-03083-8
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DOI: https://doi.org/10.1186/s13660-024-03083-8