Abstract
In this article, we introduce generalized beta extension of Sz\(\acute{a}\)sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these operators. Further, we study the uniform convergence and order of approximation in terms of Korovkin-type theorem and modulus of continuity for the space of univariate continuous functions and bivariate continuous functions in their sections.. Moreover, numerical estimates and graphical representations for convergence of one- and two-dimensional sequences of operators are studied. In continuation, local and global approximation properties are studied in terms of the first- and second-order modulus of smoothness, Peetre’s K-functional and weight functions in various functional spaces.
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1 Introduction
In 1950, Szász operators (Szász (1950)) are defined by
where \(u\in [0,\infty ),m=1,2,...,\)
The classical Sz\(\acute{a}\)sz–Mirakjan operators are linear positive operators and approximate the continuous functions over the positive semi-axis. Several mathematicians have constructed various generalizations of Sz\(\acute{a}\)sz–Mirakjan operators given by (1.1), e.g., Agrawal et al. 2014; Alotaibi 2022, 2023, ; Wafi and Rao 2018; Raiz et al. 2022, 2023; Acu and Rasa 2020; Mursaleen and Nasiruzzaman 2017; Aslan 2022; Aslan and Mursaleen 2022; Aslan 2022; Aslan and Mursaleen 2022; Rao et al. 2021; Ayman-Mursaleen 2017, 2022; Mishra et al. 2023; Mohiuddine et al. 2017; Nasiruzzaman et al. 2021, 2023; Ayman-Mursaleen and Serra-Capizzano 2022; Al-Abied et al. 2021 and Kajla et al. 2021, etc. We also refer to reader for a deep historical background (Mohiuddine 2020; Mohiuddine et al. 2022; Yadav et al. 2023; Mohiuddine et al. 2023; Nasiruzzaman et al. 2023; Mohiuddine et al. 2021; Mohiuddine and Özger 2020; Özger et al. 2022; Ansari et al. 2022; Cheng and Mohiuddine 2023).
Motivated with the above development, we define Szász–Durrmeyer type operators \(S^{\nu }_{m}:L _{B}[0,\infty ) \rightarrow \mathbb {R},\) with generalized beta function as \((L _{B}[0,\infty )\) denotes the space of bounded and Lebesgue measurable functions:
where \(C_{m, i}^{\nu }(t)=\int _{0}^{1} D_{m, i}^{\nu }(t)dt\) and \(D_{m, i}^{\nu }(t)\) is given by the formula
Now, we calculate some estimates for operators defined in equation (1.3).
Let \(e_{k}(t)=t^k\), \(k=0,1,2,3,4\). Then, in the following Lemmas, we give the some moments and estimate for the operators given by (1.3).
2 Basic Estimates
Lemma 2.1
For the operators \(S^{\nu }_{m}(.;.)\) given by (1.3), the following identities are obtained:
Proof
If \(k=0\), then
If \(k=1\), then
If \(k=2\), then
If \(k=3\), then
If \(k=4\), then
\(\square\)
Lemma 2.2
The central moments of beta Szász–Mirakjan operators using Lemma 2.1 are easily calculated as follows:
3 Rate of Convergence
Definition 3.1
The modulus smoothness for a uniformly continuous function \(\tau\) is presented as follows:
for \(\tau \in C[0, \infty ).\)
For a uniformly continuous function \(\tau\) in \(C[0,\infty )\) and \(\eta >0\), one has
Theorem 3.1
For \(S^{\nu }_{m}(.;.)\), the operators introduced by (1.3) and for each \(\tau \in C[0,\infty )\cap E\), \(S^{\nu }_{m}(\tau ;u)\longrightarrow \tau (u)\) on each compact subset of \([0,\infty )\), where \(E= \big \{\tau :u\ge 0,\dfrac{\tau (u)}{1+u^{2}}\)is convergent as \(u\longrightarrow \infty \big \}.\)
Proof
In view of Korovkin-type property (iv) of Theorem 4.1.4 in Altomare and Campiti (1994), it is sufficient to show that \(S^{\nu }_{m}(e_k;u)\longrightarrow e_k,\) for \(k=0,1,2\). Using Lemma 2.1, it is obvious \(S^{\nu }_{m}(e_{0};u)\longrightarrow e_{0}(u)\) as \(n\longrightarrow \infty\) and for \(k=1\)
Similarly, we can prove for \(k=2,\) \(S^{\nu }_{m}(e_{2};u)\longrightarrow e_{2}(u)\), which completes the proof of Theorem 3.1. \(\square\)
Theorem 3.2
(See Shisha and Bond (1968)) Let \(L:C[c,d] \longrightarrow B[c,d]\) be the positive linear operator and let \(\gamma _{u}\) be the function defined by
If \(\tau \in C_{B} ([c,d]),\) for any \(u\in [c,d]\) and \(\delta >0\), the operator L verifies:
Theorem 3.3
Let \(\tau \in C_{B}[0,\infty )\). Then, for the operator \(S^{\nu }_{m}(.;.)\) presented by (1.3), one has
Proof
In terms of Lemma 2.1, 2.2 and Theorem 3.1, one has
which prove the Theorem 3.3 choosing \(\lambda =\sqrt{S^{\nu }_{m}(A_{m}^{\nu };u)}.\) \(\square\)
4 Numerical and Graphical Analysis
In this section, we depict the convergence behaviour of the operators given by (1.3) for the function \(f(u)=\dfrac{u}{2}e^{-18u}.\) In table 1, we discuss numerical behaviour for different values of \(m= 10, 15, 25\) for the operators (1.3), with \(\nu =0\) in terms of error formula \(E^{\nu }_{m}(h; u)=|S^{\nu }_{m}(f; u)-f(u)|.\) Furthermore, graphical representation of the convergence and error of the operator (1.3) are given in Figs. 1 and 2, respectively, using \(f(u)= \dfrac{u}{2}e^{-18u}\) and \(m=10, 15, 25.\)
Here, the error approximation table 1 is given, which supports our analytical and numerical results.
5 Local Approximations
The local approximation results in \(C_{B}[0,\infty )\), which is the space of real-valued continuous and bounded functions equipped with norm, \(||h||= sup_{0\le u<\infty }|h(u)|\). For any \(h\in C_{B}[0,\infty )\) and \(\delta >0,\), Peetre K-functional is defined as follows:
where \(C^{2}_{B}[0,\infty )=\bigg \{f\in C_{B}[0;\infty ):f^{'}, f^{''}\in C_{B}[0,\infty )\bigg \}\).
By DeVore and Lorentz (1993p.177, Theorem 2.4), there is fixed real constant \(C>0.\) As a result, it exists
The modulus of smoothness of second order is denoted by \(\omega _{2}(.;.)\) and is defined as follows:
Now, for \(h\in C_{B}[0,\infty ), u\ge 0\), the auxiliary operator is taken into consideration \(\widehat{S}^{\nu }_{m}(.;.)\) as follows:
Lemma 5.1
Let \(f\in C^{2}_{B}[0,\infty )\). Then, for all \(u\ge 0\), one has
where
Proof
For the auxiliary operators are defined in the Definition (5.2), we have
By Taylor’s expansion and \(h\in C^{2}_{B}[0,\infty ),\), we have
Operating (5.2) both the sides in above equation, we have
Since,
Then, we get
Applying (5.6) and (5.7) in (5.5), we obtain
which completes the proof of Lemma 5.1. \(\square\)
Theorem 5.2
Let \(h\in C^{2}_{B}[0,\infty ).\) Then, there exists a constant \(C>0\) such that
where \(\xi _{m}(u)\) is defined by the Lemma 5.1.
Proof
For \(g\in C^{2}_{B}[0,\infty )\), \(h\in C_{B}[0,\infty )\) and in view of the definition of \(\widehat{S}^{\nu }_{m}(.;.)\), one has
With the aid of Lemma 5.1 and relation in (5.3), we get
From definition of Peetre’s K-functional
which completes the proof of Theorem 5.2.
Let \(\rho _{1}>0\) and \(\rho _{2}>0\) are two fixed real values. Then, we recall Lipschitz-type space here Özarslan and Aktŭglu (2013) as follows:
\(Lip^{\rho _{1}\rho _{2}}_{M}(\gamma ):= \bigg \{h\in C_{B}[0,\infty ):|h(t)-h(u)| \le M\dfrac{|t-u|^{\gamma }}{(t+\rho _{1}u+\rho _{2}u^{2})^{\gamma /2}}:u,t\in (0,\infty )\bigg \},\) \(M>0\) is a constant and \(0<\gamma \le 1\). \(\square\)
Theorem 5.3
Let \(h\in Lip^{\rho _{1},\rho _{2}}_{M}(\gamma )\) and \(u\in (0,\infty ).\) Then, for the operators defined by (1.3), one has
where \(\gamma \in (0,1)\) and \(\eta _{m}(u)= S^{\nu }_{m}(\xi ^{2}_{u};u).\)
Proof
For \(\gamma =1\) and \(u\in [0,\infty ),\), one has
It is obvious that
for all \(u\in [0;\infty ),\), we have
Using H\(\ddot{o}\)lder’s inequality, the Theorem 5.3 now holds for \(\gamma =1\) and \(\gamma \in (0,\infty )\). with \(q_{1}=2/\gamma\) and \(q_{2}=2/2-\gamma\), we have
Since \(\dfrac{1}{t+\rho _{1}u+\rho _{2}u^{2}}<\dfrac{1}{\rho _{1}u+\rho _{2}u^{2}}\) for all \(u\in (0,\infty )\), we have
This completes the proof of Theorem 5.3.
Now, we recall \(r^{th}\) term order Lipschitz-type maximal function suggested by Lenze (1988) as follows:
and \(r\in (0,1]\). \(\square\)
Theorem 5.4
Let \(h\in C_{B}[0,\infty )\) and \(r\in (0,1]\). Then, for all \(u\in [0,\infty )\), one has
Proof
We know that
From equation (5.9), one has
By H\(\ddot{o}\)lder’s inequality with \(q_{1}=2/r\) and \(q_{2}=2/2-r\), we have
we arrive at our the desired result. \(\square\)
6 Weighted Approximation
To establish the next result, we recall some notation from Gadziev (1976). Assume that \(B_{1+u^{2}}[0,\infty )= \left\{ h(u):|h(u)|\le M_{h}(1+u^{2})\right\}\), is weighted functional space, \(M_{h}\) is a constant that is determined by h and in \(B_{1+u^{2}}[0,\infty )\), \(u\in\) \([0,\infty ), C_{1+u^{2}}[0,\infty )\) is the space continuous functions with the norm
and
K is a constant that depends on h.
The modulus of continuity for the function h with \(a>0\) and a closed interval [0, a] is as follows:
Here, we observe that for \(h\in C_{1+u^{2}}[0,\infty )\), the modulus of continuity tends to zero.
Theorem 6.1
For \(h\in C_{1+u^{2}}[0,\infty )\) and its modulus of continuity \(\omega _{b+1}(h;\delta )\) defined on \([0,b+1]\in [0,\infty )\), we have
where
Proof
From (Ibikli and Gadjieva (1995) p. 378) for \(u\in [a,b]\) and \(v\in [0,\infty )\), we have
\(|h(v)-h(u)|\le 6M_{h}(1+b^{2})(v-u)^{2}+\left( 1+\dfrac{|v-u|}{\delta }\right) \omega _{b+1}(h;\delta ).\)
Applying both sides \(S^{\nu }_{m}(.;.),\), one has
In view of Lemma (2.2) and \(u\in [a,b]\), we get
Choosing \(\delta =\delta _{m}(b)\), we arrive at our desired result \(\square\)
Theorem 6.2
If the operators \(S^{\nu }_{m}(.;.)\) given by (1.3) from \(C^{k}_{1+u^{2}}[0;\infty )\) to \(B_{1+u^{2}}[0;\infty )\) satisfying the conditions
for \(j=0, 1, 2\) Then, for each \(h\in C^{k}_{1+u^{2}}[0,\infty )\), one has
Proof
To prove this Theorem, it is enough to show that
From Lemma 2.2, we have for \(j=0\)
For \(j=1\), it is obvious.
For \(j=2\)
This implies that \(||S^{\nu }_{m}(e_{2};.)-e_2||_{1+u^{2}}\longrightarrow 0\) as \(m\longrightarrow \infty\). Hence, we complete the proof of Theorem 6.2. \(\square\)
Theorem 6.3
Let \(h\in C^{k}_{1+u^{2}}[0,\infty )\) and \(\gamma >0\) Then, we have
Proof
For fixed number \(u_{0}>0\), one possesses
Since
\(|h(a)|\le ||h||_{1+u^{2}}(1+u^{2})\), we have
Let \(\epsilon >0\) be random real number. Then, from (3.1), there exists \(m_{1}\in \mathbb {N},\) such that
for all
for all \(m_{1}\ge m\). This implies that \(T_{2}+T_{3}<2\dfrac{||h||_{1+u^{2}}}{1+u^{2}}+\dfrac{\epsilon }{3}\).
For a large enough value of \(u_{0}\), we get \(\dfrac{||h||_{1+u^{2}}}{1+u^{2}}+\dfrac{\epsilon }{6}\),
By Theorem 6.2, there exists \(m_{2}> m\) such that
Let \(m_{3}= max\{m_{1}, m_{2}\}\). Then, joining (6.2), (6.3) and (6.4), we get
The proof of the above theorem (6.2) is complete. \(\square\)
7 Bivariate Case of Extended Beta-Type Sz\(\acute{a}\)sz–Mikjan Operator \(S^{\nu }_{m}(h;u)\)
Take \(T^2=\{(u_1, u_2):0\le u_1\le 1, 0\le u_2 \le 1\}\) and \(C(T^2)\) is the class of all continuous function on \(T^2\) equipped with norm \(||f||_{C(T^2)}=\sup _{(u_1, u_2)\in T^2}|f(u_1, u_2)|.\) Then, for all \(g\in C(T^2)\) and \(m_1, m_2\in \mathbb {N},\), we introduce a bivariate sequence as follows:
where
and
Lemma 7.1
Let \(e_{j, k}= u_{1}^{j}u_{2}^{k}.\) Then, for the operator (7.1), we get
Proof
From (2.1) and linearity, property we get
\(\square\)
8 Degree of Convergence
For any \(g\in C(\mathcal {T}^2)\) and \(\eta >0,\), the modulus of continuity of the second order is given by
with \(\mid t-u_{1}\mid \le \eta _{n_1},\; \mid s-u_{2}\mid \le \eta _{n_2}\) defined by the partial modulus of continuity as follows:
Theorem 8.1
For any \(g \in C(\mathcal {T}^2),\), we have
Proof
In order to give the proof of Theorem 8.1, generally, we use the well-known Cauchy–Schwarz inequality. Thus, we see that
If we choose \(\delta _{n_1}^2=\delta _{n_1,u_{1}}^2= S_{m_1, m_2}^{\nu } ((t-u_{1})^2;u_{1},u_{2})\) and \(\delta _{n_2}^2=\delta _{n_2,u_{2}}^2= S_{m_1, m_2}^{\nu } ((s-u_{2})^2;u_{1},u_{2}),\), then we can simply achieve our objectives. \(\square\)
Here, we analyse convergence in terms of the Lipschitz class for bivariate functions. For \(M>0\) and \(\tau ,\tau \in [0,1],\), maximal Lipschitz function space on \(E \times E \subset \mathcal {T}^2\) given by
where g is continuous and bounded on \(\mathcal {T}^2\), and
Theorem 8.2
Let \(g\in \mathcal {L}_{\tau ,\tau }(E\times E).\) Then, for any \(\tau ,\tau \in [0,1],\), there exists \(M>0\) such that
where \(\delta _{n_1,u_{1}}\) and \(\delta _{n_2,u_{2}}\) defined by Theorem 8.1.
Proof
Consider \(\mid u_{1}-x_0 \mid =d(u_{1}, E)\) and \(\mid u_{2}-y_0 \mid =d(u_{2}, E)\), for any \((u_{1},u_{2})\in \mathcal {T}^2\), and \((x_0,y_0)\in E\times E.\)
Let \(d(u_{1}, E)=\inf \{ \mid u_{1}-u_{2} \mid : u_{2} \in E \}\). Then, we write here
Apply \(S_{m_1, m_2}^{\nu }(.;.,.)\), we obtain
For all \(A,B \ge 0\) and \(\tau \in [0,1],\), the inequality \((A+B)^{\tau }\le A^{\tau }+B^{\tau },\) thus
Therefore,
Apply Hölders inequality on \(S_{m_1, m_2}^{\nu }(.;.,.)\), we get
Thus, we can obtain
which completes the proof. \(\square\)
9 Numerical and Graphical Analysis
It is observed by given below example, table and figure for the different set of parameters \(\nu =0.3\), the operator \(S_{m_1, m_2}^\nu (.;.,.)\) converges uniformly to the function \(f(u_{1}, u_{2})=\dfrac{u_1u_2}{2}e^{-16(u_{1}u_{2})}\) (Yellow), and using error formula \(E_{m_1, m_2}^{\nu }(f; u_1, u_2)=|S_{m_1, m_2}^{\nu }(h; u_{1}, u_{2})-f(u_{1}, u_{2})|\), the error of the operator (7.1) is given in Fig. 4. Further, as \(m_{1}=m_{2}=10\) (Blue), \(m_{1}=m_{2}=15\) (Green) and \(m_{1}=m_{2}=25\) (Red).
10 Conclusion
In this study, we introduce generalized beta Sz\(\acute{a}\)sz–Mirakjan operators and study their approximation properties. Further, we prove a Korovkin-type convergence theorem, the order of convergence concerning the usual modulus of continuity as well as Peetre’s K-functional and Lipschitz-type class of functions. Moreover, we introduced global approximation results and A-Statistical approximation properties of the constructed operators. We provided several graphical representations as convergence and error of approximation in terms of the values of some selected parameters in order to make our article comprehensible and to demonstrate the accuracy and efficacy of the proposed operators.
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Rao, N., Raiz, M., Ayman-Mursaleen, M. et al. Approximation Properties of Extended Beta-Type Szász–Mirakjan Operators. Iran J Sci 47, 1771–1781 (2023). https://doi.org/10.1007/s40995-023-01550-3
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DOI: https://doi.org/10.1007/s40995-023-01550-3