Abstract
The aim of this paper is to study a new generalization of Lupaş-type operators whose construction depends on a real-valued function ρ by using two sequences \(u_{m} \) and \(v_{m}\) of functions. We prove that the new operators provide better weighted uniform approximation over \([0,\infty )\). In terms of weighted moduli of smoothness, we obtain degrees of approximation associated with the function ρ. Also, we prove Voronovskaya-type theorem, quantitative estimates for the local approximation.
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1 Introduction
The Weierstrass approximation theorem is the basis of approximation theory introduced by Weierstrass [15], which states that each continuous function defined on \([a,b]\) can be approximated uniformly by some polynomial. In 1912, Bernstein [3] established a constructive proof of the Weierstrass theorem by using Korovkin’s theorem [9].
On the other hand, Cárdenas et al. [4] defined the Bernstein-type operators by \(B_{m}(go\rho ^{-1})o\rho \) and also presented a better degree of approximation depending on ρ. This type of approximation operators generalizes the Korovkin set from \(\{e_{0}, e_{1}, e_{2}\}\) to \(\lbrace e_{0}, \rho , \rho ^{2} \rbrace \). In 2014, Aral et al. [1] also proposed a new modification of Szász–Mirakyan type operators to investigate approximation properties of the announced operators acting on functions defined on unbounded intervals \([0,\infty )\). For various other generalizations of Szász–Mirakyan type operators, one can go through these papers [12–14] of Srivastava et al.
Very recently, for \(m\geq 1\), \(z\geq 0\), and suitable functions g defined on \([0,\infty )\), Hatice et al. [8] introduced a new modification of Lupaş operators [11] using a suitable function ρ as follows:
where ρ satisfies following properties:
- \((\rho _{1})\):
-
ρ is a continuously differentiable function on \([0,\infty )\),
- \((\rho _{2})\):
-
\(\rho (0)=0\) and \(\inf_{z\in [0,\infty )}\rho ^{\prime }(z)\geq 1\), and \((m\rho (z))_{j}\) is the rising factorial defined as follows:
$$\begin{aligned}& \bigl(m\rho (z)\bigr)_{0}=1, \\& \bigl(m\rho (z)\bigr)_{j}=\bigl(m\rho (z)\bigr) \bigl(m\rho (z)+1 \bigr) \bigl(m\rho (z)+2\bigr)\cdots \bigl(m\rho (z)+j-1\bigr), \quad j \geq 0. \end{aligned}$$
If we put \(\rho (z)=z\) in (1.1), then it reduces to the classical Lupaş operators defined in [11].
Very recently, a new construction of Szász–Mirakjan operators was given by Aral et al. [2] by using ρ and two sequences of functions \(\alpha _{m}\), \(\beta _{m}\) defined on a subinterval of \([0,\infty )\):
Inspired by the idea used by Aral et al. in [2], in this paper we define a new construction of Lupaş operator (1.1) which depends on \(\alpha _{m}(z)\) and \(\beta _{m}(z)\), where \(\alpha _{m}(z)\) and \(\beta _{m}(z)\) are sequences of functions defined on \(\tilde{E}\subset [0,\infty )\).
The paper is organized as follows. In Sect. 2, the construction of the announced operator is presented and its moments and central moments are calculated. In Sect. 3, we study convergence properties by using weighted space. In Sect. 4, we obtain the rate of convergence of new constructed operators associated with the weighted modulus of continuity. In Sect. 5, we prove Voronovskaya-type asymptotic formula. Finally, in Sect. 6, we give some approximation results related to \(\mathcal{K}\)-functional, also we define a Lipschitz-type functions.
2 The construction of Lupaş-type operators
Let g be a continuous functions on \([0,\infty )\) and \(\tilde{E}\subset [0,\infty )\). For given \(m_{0}\in \mathbb{N}\), define \(\mathbb{N}_{1}=\{m\in \mathbb{N}:m\geq m_{0}\}\).
Let \(\alpha _{m},\beta _{m}:\tilde{E}\rightarrow \mathbb{R}\) such that
where \(\alpha _{m}\), \(\beta _{m}\) are positive functions defined on Ẽ.
Then we consider the new operators in the following form:
where ρ is a function which satisfies the conditions \((\rho _{1})\) and \((\rho _{2})\).
We will impose some assumptions on these operators, to show the sequence of operators (2.2) is an approximation process.
We suppose that, for \(z\in \tilde{E}\),
where \(u_{m}:\tilde{E}\rightarrow \mathbb{R}\). From (2.2), we obtain
Thus, we get
Secondly, we assume that
where \(v_{m}:\tilde{E}\rightarrow \mathbb{R}\). From (2.2), we infer
From (2.4) and (2.5), we obtain
Now combining (2.3) and (2.6), we get
and from relation (2.3), we can write
and
where \(u_{m}(z)>-1\) for any \(z\in \tilde{E}\) and \(m\in \mathbb{N}_{1}\).
Therefore, as a consequence, operators (2.2) become
for \(m\in \mathbb{N}_{1}\) and for any \(z\in \tilde{E}\).
We can recover some linear positive operators which are already in the literature. From operators (2.9) and for the particular choices of the functions \(u_{m}\), \(v_{m}\), and ρ:
-
(i)
If we take \(u_{m}(z)=v_{m}(z)=0\), operators (2.9) turn out to be operators (1.1).
-
(ii)
If we take \(u_{m}(z)=v_{m}(z)=0\), \(\rho (z)=z\), operators (2.9) turn out to be the classical Lupaş operators given in [11] by
$$ L_{m}(g;z)=2^{-mz}\sum_{j=0}^{\infty } \frac{(mz)_{j}}{2^{j}j!}g \biggl( \frac{j}{m} \biggr). $$
Now, in order to obtain weighted approximation processes, we assume that the following inequalities hold:
such that
From (2.10) and (2.11) it is clear that \((L^{\rho }_{m}(g;z) )_{m\geq m_{0}}\) is an approximation process on \(\tilde{E}\subset [0,\infty )\).
Now, we give some lemmas which are required to prove our main results.
Lemma 2.1
For the operators \(L^{\rho }_{m}(g;z)\) and for all \(z\in \tilde{E}\), we have:
-
1.
\(L^{\rho }_{m}(1;z)=1+ u_{m}(z)\),
-
2.
\(L^{\rho }_{m}(\rho ;z)=\rho (z) + v_{m}(z)\),
-
3.
\(L^{\rho }_{m}(\rho ^{2};z)= \frac{ (\rho (z) + v_{m}(z) )^{2}}{1+ u_{m}(z)}+ \frac{2}{m}(\rho (z) + v_{m}(z))\),
-
4.
\(L^{\rho }_{m}(\rho ^{3};z)= \frac{ (\rho (z) + v_{m}(z) )^{3}}{ ( 1+ u_{m}(z) )^{2}}+ \frac{6 (\rho (z) + v_{m}(z) )^{2}}{m (1+ u_{m}(z) )}+ \frac{6}{m^{2}}(\rho (z) + v_{m}(z))\),
-
5.
\(L^{\rho }_{m}(\rho ^{4};z)= \frac{ (\rho (z) + v_{m}(z) )^{4}}{ ( 1+ u_{m}(z) )^{3}}+ \frac{12 (\rho (z) + v_{m}(z) )^{3}}{m ( 1+ u_{m}(z) )^{2}}+ \frac{36 (\rho (z) + v_{m}(z) )^{2}}{m^{2} (1+ u_{m}(z) )}+ \frac{26}{m^{3}}(\rho (z) + v_{m}(z))\).
Lemma 2.2
By using Lemma 2.1and by the linearity of operators \(L^{\rho }_{m}\), we can acquire the central moments as follows:
-
1.
\(L^{\rho }_{m}(\rho (\zeta )-\rho (z);z) = v_{m}(z)-\rho (z)u_{m}(z)\),
-
2.
\(L^{\rho }_{m}{((\rho (\zeta )-\rho (u))^{2};u)}= \frac{ (\rho (z) + v_{m}(z) )^{2}}{1+ u_{m}(z)}+ \frac{2 (\rho (z) + v_{m}(z) ) ( 1-m\rho (z) )}{m}+ (1 + u_{m}(z) )\rho ^{2}(z)\),
-
3.
\(L^{\rho }_{m}{((\rho (\zeta )-\rho (u))^{3};u)}= \frac{ (\rho (z) + v_{m}(z) )^{3}}{ ( 1+ u_{m}(z) )^{2}}+ \frac{3 (\rho (z) + v_{m}(z) )^{2} ( 2-n\rho (z) )}{m(1 + u_{m}(z))}+ \frac{ (\rho (z) + v_{m}(z) ) ( 6-6n\rho (z)+3m^{2}\rho ^{2}(z) )}{m^{2}}- (1 + u_{m}(z) )\rho ^{3}(z)\),
-
4.
\(L^{\rho }_{m}{((\rho (\zeta )-\rho (u))^{4};u)}= \frac{ (\rho (z) + v_{m}(z) )^{4}}{ ( 1+ u_{m}(z) )^{3}}+ \frac{ (\rho (z) + v_{m}(z) )^{3} ( 12-4n\rho (z) )}{m(1 + u_{m}(z))^{2}}+ \frac{ (\rho (z) + v_{m}(z) )^{2} ( 36-24n\rho (z)+6m^{2}\rho ^{2}(z) )}{m^{2}(1 + u_{m}(z))}+ \frac{ (\rho (z) + v_{m}(z) ) ( 26-24n\rho (z)+12m^{2}\rho ^{2}(z)-4m^{3}\rho ^{3}(z) )}{m^{3}}+ (1 + u_{m}(z) )\rho ^{4}(z)\).
Remark 2.3
If we put \(u_{m}(z)=v_{m}(z)=0\) in Lemma 2.1 and Lemma 2.2, we have the same result proved in Lemma 2 and Lemma 3 of [8].
3 Direct result in weighted space
In this section, by using weighted space, we discuss some convergence properties for the operators \(L^{\rho }_{m}\).
Let \(\Phi (z)=1+\rho ^{2}(z)\) be a weight function and \(\mathcal{B}_{\Phi }[0,\infty )\) be the weighted spaces defined as follows:
where \(\mathcal{K}_{g}\) is a constant and \(\mathcal{B}_{\Phi }[0,\infty )\) is a normed linear space equipped with the norm
Also, the subspaces \(\mathcal{C}_{\Phi }[0,\infty )\), \(U_{\Phi }[0,\infty )\) and \(U_{\Phi }[0,\infty ) \) of \(\mathcal{B}_{\Phi }[0,\infty )\) are defined as
It is obvious that \(\mathcal{C}^{*}_{\Phi }[0,\infty )\subset U_{\Phi }[0,\infty )\subset \mathcal{C}_{\Phi }[0,\infty )\subset \mathcal{B}_{\Phi }[0,\infty )\).
In [6], Gadjiev proved the following results for the weighted Korovkin-type theorems.
Lemma 3.1
([6])
For \({m\geq 1}\), \(\mathcal{G}_{m}: \mathcal{B}_{\Phi }[0,\infty )\rightarrow \mathcal{B}_{\Phi }[0,\infty )\) satisfying
holds, where \(\mathcal{K}_{m}>0\) is a constant depending on m.
Theorem 3.2
([6])
For \({m\geq 1}\), \(\mathcal{G}_{m}: \mathcal{B}_{\Phi }[0,\infty )\rightarrow \mathcal{B}_{\Phi }[0,\infty )\) satisfies
Then, for any function \(g\in C^{*}_{\Phi }[0,\infty )\), we obtain
Therefore, our result follows.
Theorem 3.3
For each \(g\in C^{*}_{\Phi }[0,\infty )\), the following relation
holds, provided that conditions (2.10) and (2.11) are fulfilled.
Proof
Let \(g\in C^{*}_{\Phi }[0,\infty )\). Then \(|g(z)|\leq \mathcal{H}_{g}\Phi (z)\), \(z\geq 0\). \(L^{\rho }_{m}\) being linear and positive, it is monotone. Thus
implies that the operator \(L^{\rho }_{m}\) maps the space \(C_{\Phi }[0,\infty )\) into \(B_{\Phi }[0,\infty )\).
By (2.10) and Lemma 2.1, we have
As we know each \(L^{\rho }_{m}(g;z)\) is defined on Ẽ. Now, by considering the following sequence of operators, we extend it on \([0,\infty )\):
Obviously,
By applying 3.2 to the operators \(\mathcal{G}_{m}=\mathcal{A}_{m}\) the claim will be proved.
Hence, we have to prove that
Since
by using (3.1), we have
By using (3.2), we get the desired result. □
4 Rate of convergence
In this section, by using weighted modulus of continuity \(\omega _{\rho }(g;\delta )\), we determine the rate of convergence for \(L^{\rho }_{m}\) which was recently considered by Holhoş [7] as follows:
where \(g\in \mathcal{C}_{\Phi }[0,\infty )\), with the following properties:
-
(i)
\(\omega _{\rho }(g;0)=0\),
-
(ii)
\(\omega _{\rho }(g;\delta )\geq 0\), \(\delta \geq 0\) for \(g\in \mathcal{C}_{\Phi }[0,\infty )\),
-
(iii)
\(\lim_{\delta \rightarrow 0}\omega _{\rho }(g;\delta )=0 \) for each \(g\in U_{\Phi }[0,\infty )\).
Theorem 4.1
([7])
Let us consider a sequence of positive linear operators \(\mathcal{G}_{m}:\mathcal{C}_{\Phi }[0,\infty )\rightarrow \mathcal{B}_{ \Phi }[0,\infty )\) with
where the sequences \(a_{m}\), \(b_{m}\), \(c_{m}\), and \(d_{m}\) converge to zero as \(m\rightarrow \infty \). Then
for all \(g\in \mathcal{C}_{\Phi }[0,\infty )\), where
Theorem 4.2
Let us have, for each \(g\in \mathcal{C}_{\Phi }[0,\infty )\),
where
Proof
If we calculate the sequences \((a_{m})\), \((b_{m})\), \((c_{m})\), and \((d_{m})\), then by using Lemma 2.1, clearly we have
and
Finally,
Thus conditions (4.1)–(4.5) are satisfied. Now, by Theorem 4.1, we obtain the desired result. □
Remark 4.3
From property \((iii)\) of \(\omega _{\rho }(g;\delta )\) and Theorem 4.2, we have
5 Voronovskaya-type theorem
In this section, we establish Voronovskaya-type result for \(L^{\rho }_{m}\).
Theorem 5.1
Let \(g\in \mathcal{C}_{\Phi }[0,\infty )\), \(z\in [0,\infty )\) and suppose that \((go\rho ^{-1} )^{\prime }\) and \((go\rho ^{-1} )^{\prime \prime }\) exist at \(\rho (z)\). If \((go\rho ^{-1} )^{\prime \prime }\) is bounded on \([0,\infty )\) and
then we have
Proof
By using the Taylor expansion of \((go\rho ^{-1} )\) at \(\rho (z)\in [0,\infty )\), there exists a point ζ lying between z and z, we have
where
Therefore, the assumption on g and (5.2) ensure that
and \(\lim_{\zeta \rightarrow z}\lambda _{z}(\zeta )=0\). By applying operators (2.9) to (5.1), we obtain
From Lemmas 2.1 and 2.2, we obtain
and
Since from (5.2), for every \(\epsilon >0\), \(\lim_{\zeta \rightarrow z}\lambda _{z}(\zeta )=0\). Let \(\delta >0\) such that \(|\lambda _{z}(\zeta )|<\epsilon \) for every \(\zeta \geq 0\). From the Cauchy–Schwarz inequality, we get immediately
Since
we obtain
Thus, taking into account equations (5.4), (5.5), and (5.7) to equation (5.3), this proves the theorem. □
6 Local and global approximation
In order to prove local approximation theorems for the operators, let \(\mathcal{C}_{B}[0,\infty )\) be the space of real-valued continuous and bounded functions g with the norm \(\| \cdot \| \) given by
We begin by considering the \(\mathcal{K}\)-functional
where \(\delta >0\) and \(W^{2}=\{s\in \mathcal{C}_{B}[0,\infty ):r^{{\prime }},r^{{\prime \prime }}\in \mathcal{C}_{B}[0,\infty )\}\). Then, in view of the known result [5], there exists an absolute constant \(\mathcal{D}>0\) such that
For \(g\in \mathcal{C}_{B} [0,\infty )\), the second order modulus of smoothness is defined as
and the usual modulus of continuity is defined as
Theorem 6.1
There exists an absolute constant \(\mathcal{D}>0\) such that
where \(g\in \mathcal{C}_{B} [0,\infty )\) and
Proof
Let \(r\in W^{2}\) and \(z,\zeta \in [0,\infty )\). By using Taylor’s formula we have
By using the equality
putting \(v= \rho (z)\) in the last term in equality (6.2), we get
By applying \(\mathcal{S^{*}}^{\mu ,\lambda }_{m,\rho }\) to (6.2) and also by using Lemma 2.1 and (6.4), we deduce
By using conditions (\(\rho _{1}\)) and (\(\rho _{2}\)), we get
where
For all \(g\in \mathcal{C}_{B} [0,\infty )\), we have
Hence we have
If we choose \(\mathcal{D}=\operatorname{max} \{2, \|\rho ^{\prime \prime }\|\}\), then
Taking infimum over all \(r\in W^{2}\), we obtain
□
Let \(0<\alpha \leq 1\), ρ be a function with conditions (\(\rho _{1}\)), (\(\rho _{2}\)) and \(\mathrm{Lip}_{\mathcal{M}}(\rho (y); \alpha )\), \(\mathcal{H}\geq 0\) satisfying
Moreover, \(\mathcal{Y}\subset [0, \infty )\) is a bounded subset and the function \(g\in \mathrm{Lip}_{\mathcal{M}}(\rho (z); \alpha )\), \(0<\alpha \leq 1\) on \(\mathcal{Y}\) if
where \(\mathcal{H}_{\alpha , g}\) is a constant.
Theorem 6.2
Let ρ be a function satisfying conditions (\(\rho _{1}\)), (\(\rho _{2}\)). Then, for any \(g\in \mathrm{Lip}_{\mathcal{H}}(\rho (z); \alpha )\), \(0<\alpha \leq 1\) and for every \(z\in (0,\infty )\), \(m\in \mathbb{N}\), we have
where
Proof
Assume that \(\alpha =1\). Then, for \(g\in \mathrm{Lip}_{\mathcal{M}}(\alpha ; 1)\) and \(z\in (0,\infty )\), we have
By applying the Cauchy–Schwarz inequality, we get
Let us assume that \(\alpha \in (0,1)\). Then, for \(g\in \mathrm{Lip}_{\mathcal{H}}(\alpha ; 1)\) and \(z\in (0,\infty )\), we have
By taking \(p=\frac{1}{\alpha }\) and \(q=\frac{1}{1-{\alpha }}\), \(g\in \mathrm{Lip}_{\mathcal{H}}(\rho (z); \alpha )\), and applying Hölder’s inequality, we have
Finally, by applying the Cauchy–Schwarz inequality, we get
□
Theorem 6.3
Let ρ be a function satisfying conditions (\(\rho _{1}\)), (\(\rho _{2}\)) and \(\mathcal{Y}\) be a bounded subset of \([0,\infty )\). Then, for any \(g\in \mathrm{Lip}_{\mathcal{H}}(\rho (z); \alpha )\), \(0<\alpha \leq 1\), on \(\mathcal{Y}\) \(\alpha \in (0,1]\), we have
where \(d(z,\mathcal{Y})= \inf \{\|z-x\|: x \in \mathcal{Y}\}\) and \(\mathcal{H}_{\alpha , g}\) is a constant depending on α and g, and
Proof
Let \(\overline{\mathcal{Y}}\) be the closure of \(\mathcal{Y}\) in \([0,\infty )\). Then there exists a point \({z_{0}} \in \overline{\mathcal{Y}}\) such that \(d(z,\mathcal{Y})= |z-{z_{0}}|\).
Using the monotonicity of \(L^{\rho }_{m}\) and the hypothesis of g, we obtain
By using Hölder’s inequality for \(p=\frac{2}{\alpha }\) and \(q=\frac{2}{2-{\alpha }}\), as well as the fact \(|\rho (z)-\rho (z_{0})|=\rho ^{\prime }(z)|\rho (z)-\rho (z_{0})|\) in the last inequality, we get
Hence, by Lemma 2.2 we get the proof. □
Now, we recall the local approximation given in [10] for \(g\in \mathcal{C}_{B} [0,\infty )\) as follows:
Then we get the next result.
Theorem 6.4
Let \(g\in \mathcal{C}_{B}[0,\infty )\) and \({\alpha }\in (0,1]\). Then, for all \(z\in [0,\infty )\), we have
where
Proof
We know that
From equation (6.7), we have
By applying Hölder’s inequality with \(p=\frac{2}{\alpha }\) and \(q=\frac{2}{2-{\alpha }}\), we have
which proves the desired result. □
Conclusion. Here, a new construction of the generalized Lupaş operators is constructed. We have investigated convergence properties, order of approximation, Voronovskaja-type results, and quantitative estimates for the local approximation. The constructed operators have better flexibility and rate of convergence which depend on the selection of the function ρ and the sequences \(u_{m}\), \(v_{m}\).
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Acknowledgements
The first author(Mohd Qasim) is grateful to the Council of Scientific and Industrial Research (CSIR), India, for providing the Senior Research Fellowship with file no. (09/1172(0001)/2017-EMR-I). Also, the authors thank Fujian Provincial Key Laboratory of Data-Intensive Computing, Fujian University Laboratory of Intelligent Computing and Information Processing and Fujian Provincial Big Data Research Institute of Intelligent Manufacturing of China.
Funding
This work is supported by the Natural Science Foundation of Fujian Province of China (Grant No. 2020J01783), the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant No. 2018C087R), and the Program for New Century Excellent Talents in Fujian Province University.
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Qasim, M., Khan, A., Abbas, Z. et al. A new construction of Lupaş operators and its approximation properties. Adv Differ Equ 2021, 51 (2021). https://doi.org/10.1186/s13662-020-03143-5
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DOI: https://doi.org/10.1186/s13662-020-03143-5