Abstract
In the present paper, we introduce the Chlodowsky variant of \((p,q)\) Bernstein-Stancu-Schurer operators which is a generalization of \((p,q)\) Bernstein-Stancu-Schurer operators. We also discuss its Korovkin-type approximation properties and rate of convergence.
Similar content being viewed by others
1 Introduction and preliminaries
In 1912, Bernstein [1] introduced the following sequence of operators \(B_{n} : C[0, 1] \rightarrow C[0, 1]\) defined for any \(n \in \mathbb{N}\) and for any function \(f \in C[0, 1]\):
Later various generalizations of these operators were discovered. It has been proved as a powerful tool for numerical analysis, computer aided geometric design and solutions of differential equations. In last two decades, the applications of q-calculus has played an important role in the area of approximation theory, number theory and theoretical physics. In 1987, Lupaş [2] and in 1997, Phillips [3] introduced a sequence of Bernstein polynomials based on q-integers and investigated its approximation properties. Several researchers obtained various other generalizations of operators based on q-calculus. For any function \(f \in C[0, 1]\) the q-form of Bernstein operator is described by Lupaş [2] as
In 1932, Chlodowsky [4] presented a generalization of Bernstein polynomials on an unbounded set, known as Bernstein-Chlodowsky polynomials,
where \(b_{n}\) is an increasing sequence of positive terms with the properties \(b_{n} \rightarrow\infty\) and \(\frac{b_{n}}{n} \rightarrow0\) as \(n \rightarrow\infty\).
In 2008, Karsli and Gupta [5] expressed the q-analogue of Bernstein-Chlodowsky polynomials by
where \(b_{n}\) is an increasing sequence of positive values, with the properties \(b_{n} \rightarrow\infty\) and \(\frac{b_{n}}{[n]_{q}} \rightarrow 0\) as \(n \rightarrow\infty\).
Recently, Mursaleen et al. [6–9] proposed and analyzed approximation properties for \((p,q)\) analogue of Bernstein operators, Bernstein-Stancu operators and Bernstein-Schurer operators. Besides this, we also refer to some recent related work on this topic: e.g. [10–20].
In 2015, Mursleen et al. [7], investigated the \((p,q)\) form of the Bernstein-Stancu operator, which is given by
where \(\alpha, \beta\) are non-negative integers and \(f \in C[0, 1]\), \(x \in[0,1]\) and \(0 \leq\alpha\leq\beta\).
For the first few moments, we get the following lemma.
Lemma 1
See [7]
For the operators \(S_{n}^{(\alpha,\beta)}\), we have
-
1.
\(S_{n}^{(\alpha,\beta)}(1;x,p,q)=1\),
-
2.
\(S_{n}^{(\alpha,\beta)}(t;x,p,q)=\frac{[n]_{p,q}x+\alpha }{[n]_{p,q}+\beta}\),
-
3.
\(S_{n}^{(\alpha,\beta)}(t^{2};x,p,q)=\frac{1}{([n]_{p,q}+\beta )^{2}}(q[n]_{p,q}[n-1]_{p,q}x^{2}+[n]_{p,q}(2\alpha+ p^{n-1})x+\alpha^{2})\).
2 Construction of the operators
Considering the revised form of \((p,q)\) analogue of Bernstein operators [7], we construct the Chlodowsky variant of \((p,q)\) Bernstein-Stancu-Schurer operators as
where \(n\in\mathbb{N}\), \(m,\alpha,\beta\in\mathbb{N}_{0}\), with \(\frac {\alpha}{\beta} \approx1\), \(0 \leq x \leq b_{n}\), \(0< q < p\leq1 \) and \(b_{n}\) is an increasing sequence of positive terms with the properties \(b_{n} \rightarrow\infty\) and \(\frac{b_{n}}{[n]_{p,q}} \to0\) as \(n \to\infty\). Evidently, \(C_{n,m}^{(\alpha,\beta)}\) is a linear and positive operator. Consider the case if \(p,q \rightarrow1\) and \(m= 0\) in (2.1), then it will reduce to the Stancu-Chlodowsky polynomials [21].
Let us assume the number \(n+m=n_{m}\), we will use this notation throughout in this paper. Next, we have obtained the following lemma using simple calculations.
Lemma 2
Let \(C_{n,m}^{(\alpha,\beta)}(f;x,p,q)\) be given by (2.1). The first few moments of the operators are
-
(i)
\(C_{n,m}^{(\alpha,\beta)}(1;x,p,q)=1\),
-
(ii)
\(C_{n,m}^{(\alpha,\beta)}(t;x,p,q)=\frac {[n_{m}]_{p,q}x+\alpha b_{n}}{[n]_{p,q}+\beta}\),
-
(iii)
\(C_{n,m}^{(\alpha,\beta)}(t^{2};x,p,q)=\frac {1}{([n]_{p,q}+\beta )^{2}}(q[n_{m}]_{p,q}[n_{m}-1]_{p,q}x^{2}+[n_{m}]_{p,q}(2\alpha+p^{n_{m}-1})b_{n} x+\alpha^{2} b_{n}^{2})\),
-
(iv)
\(C_{n,m}^{(\alpha,\beta)}((t-x);x,p,q)= (\frac {[n_{m}]_{p,q}}{[n]_{p,q}+\beta}-1 )x+\frac{\alpha b_{n}}{[n]_{p,q}+\beta}\),
-
(v)
$$\begin{aligned} C_{n,m}^{(\alpha,\beta)}\bigl((t-x)^{2};x,p,q\bigr) ={}& \biggl(1-2\frac {[n_{m}]_{p,q}}{([n]_{p,q}+\beta)}+\frac {q[n_{m}]_{p,q}[n_{m}-1]_{p,q}}{([n]_{p,q}+\beta)^{2}} \biggr)x^{2} \\ &+ \biggl( \frac{(2\alpha+p^{n_{m}-1})[n_{m}]_{p,q}}{([n]_{p,q}+\beta )}-2\alpha \biggr) \frac{b_{n}}{([n]_{p,q}+\beta)} x\\ & + \frac{\alpha^{2} b_{n}^{2}}{([n]_{p,q}+\beta)^{2}}. \end{aligned}$$
Proof
(i)
(ii)
(iii)
Now using \([k+1]_{p,q}=p^{k}+q[k]_{p,q}\), we will obtain the result.
Using the linear property of operators, we have
Hence, we get (iv).
Similar calculations give
Substituting the results of (i), (ii) and (iii), we prove the result (v). □
Lemma 3
For every fixed \(0< q< p \leq1\), we have
Proof
Thus, \([n_{m}]_{p,q}[n_{m}-1]_{p,q} q \leq[n_{m}]_{p,q}^{2}\), and we get
We can conclude the last inequality using the following statements:
Since \(0 < q < p \leq1\), we have \(0 < q^{n} < p^{n} \leq1\) and \(0<(1-p^{m})<(1-q^{m}) \leq1\), hence \(q^{n}(1-p^{m})< p^{n}(1-q^{m})\) i.e. \(p^{n}(1-q^{m})-q^{n}(1-p^{m})>0\). □
Remark 1
As a result of Lemma 2 and 3, we have
3 Results and discussion
In this paper we have constructed and investigated a Chlodowsky variant of \((p,q)\) Bernstein-Stancu-Schurer operator. We have showed that our modified operators have a better error estimation than the classical ones. We have also obtained some approximation results with the help of the well-known Korovkin theorem and the weighted Korovkin theorem for these operators. Furthermore, we studied convergence properties in terms of the modulus of continuity for functions in Lipschitz class. Next we have also obtained the Voronovskaja-type result for these operators.
3.1 Korovkin-type approximation theorem
Assume \(C_{\rho}\) is the space of all continuous functions f such that
and \(\rho(x)\) is the weight function.
Then \(C_{\rho}\) is a Banach space with the norm
Consider the subspace \(C_{\rho}^{0} := \{ f \in C_{\rho}: \lim_{|x| \to\infty} \frac{|f(x)|}{\rho(x)}\text{ is finite}\}\).
The subsequent Theorem 1 is a Korovkin approximation theorem in weighted space.
Theorem 1
See [22]
There exists a sequence of positive linear operators \(U_{n}\), acting from \(C_{\rho}^{0}\) to \(C_{\rho}^{0} \), satisfying the conditions
-
(1)
\(\lim_{n\rightarrow\infty}\|U_{n}(1;\cdot)-1\|_{\rho}=0\),
-
(2)
\(\lim_{n\rightarrow\infty}\|U_{n}(\phi;\cdot)-\phi\| _{\rho}=0\),
-
(3)
\(\lim_{n\rightarrow\infty}\|U_{n}(\phi^{2};\cdot)-\phi^{2}\| _{\rho}=0\),
where \(\phi(x)\) is a continuous and increasing function on \((-\infty ,\infty)\) such that \(\lim_{x\to\pm\infty} \phi(x) = \pm\infty \) and \(\rho(x) = 1 + \phi^{2}\), and there exists a function \(f^{*} \in C_{\rho}^{0}\) for which
Consider the weight function \(\rho(x)=1+x^{2}\) and operator (see [23])
For \(f \in C_{1+x^{2}}\), we have
Now, using Lemma 2 we will obtain
which means that \(U_{n,m}^{\alpha,\beta}(f;\cdot,p,q)\) is bounded operator, henceforth a continuous operator too. Since ‘An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator.’
Now, consider the sequences \((p_{n})\) and \((q_{n})\) for \(0< q_{n}< p_{n}\leq1\) satisfying
Theorem 2
For all \(f\in C_{1+x^{2}}^{0}\), \(0\leq x \leq b_{n}\), we have
provided that \(p:=(p)_{n}\), \(q:=(q)_{n}\) with \(0< q_{n} < p_{n}\leq1\) satisfying (3.2) and \(\lim_{n\to\infty}\frac{b_{n}}{[n]_{p_{n},q_{n}}}=0\).
Proof
Using the results of Theorem 1 and Lemma 2(i), (ii) and (iii), we will obtain the following assessments, respectively:
and
whenever \(n \to\infty\).
Since the weight function is invariant w.r.t. positive and negative values of x, and conditions (3.4)-(3.6) are true for all \(t\in\mathbb{R}\), we can use Theorem 1 and get the desired result (3.3), which implies that the operator sequence \(C_{n,m}^{\alpha,\beta}\) converges uniformly to any continuous function in weighted space \(C_{1+x^{2}}^{0}\) for \(x \in[0, b_{n}]\). □
Theorem 3
Assuming c as a positive and real number independent of n and f as a continuous function which vanishes on \([c,\infty)\). Let \(p := (p_{n})\), \(q := (q_{n})\) with \(0 < q_{n} < p_{n} \leq1\) satisfying (3.2) and \(\lim_{n\to\infty} \frac{b_{n}^{2}}{[n]_{p_{n},q_{n}}} =0\). Then we have
Proof
From the hypothesis on f, it is bounded i.e. \(|f(x)| \leq M\) (\(M>0\)). For any \(\epsilon>0\), we have
where \(x\in[0,b_{n}]\) and \(\delta=\delta(\epsilon)\) are independent of n. Operating with the operator (2.1) on both sides, we can conclude by using Lemma 3 and Remark 1,
Since \(\frac{b_{n}^{2}}{[n]_{p_{n},q_{n}}} =0\) as \(n \to\infty\), we have the desired result. □
3.2 Rate of convergence
We will find the rate of convergence for functions in the Lipschitz class \(\mathit{Lip}_{M}(\gamma)\) (\(0 < \gamma\leq1\)). Assume that \(C_{B}[0,\infty )\) denotes the space of bounded continuous functions on \([0,\infty)\). A function \(f \in C_{B}[0,\infty)\) belongs to \(\mathit{Lip}_{M}(\gamma)\) if
Theorem 4
Let \(f \in \mathit{Lip}_{M}(\gamma)\), then
where \(\lambda_{n,p,q}(x)=C_{n,m}^{\alpha,\beta}((t-x)^{2};x,p,q)\).
Proof
Since \(f \in \mathit{Lip}_{M}(\gamma)\), and the operator \(C_{n,m}^{\alpha,\beta }(f;x,p,q)\) is linear and monotone,
Using Hölder’s inequality with the values \(p=\frac{2}{\gamma}\) and \(q=\frac{2}{2-\gamma}\), we get
□
In order to obtain rate of convergence in terms of modulus of continuity \(\omega(f;\delta)\), we assume that, for any \(f \in C_{B}[0,\infty)\) and \(x \geq0\), the modulus of continuity of f is given by
Thus it implies for any \(\delta> 0\)
Theorem 5
If \(f \in C_{B}[0,\infty)\), we have
where \(\omega(f;\cdot)\) is the modulus of continuity of f and \(\lambda _{n,p,q}(x)\) is the same as in Theorem 4.
Proof
Using the triangular inequality, we get
Now using (3.8) and Hölder’s inequality, we get
Now choosing \(\delta=\lambda_{n,p,q}(x)\) as in Theorem 4, we have
□
Next we calculate the rate of convergence in terms of the modulus of continuity of the derivative of a function.
Theorem 6
Let \(A>0\). If \(f(x)\) has a continuous bounded derivative \(f {'}(x)\) and \(\omega(f{'} ; \delta)\) is the modulus of continuity of \(f{'}(x)\) in \(x\in[0,\max\{b_{n},A\}]\), then
where M is a positive constant such that \(|f{'}(x)| \leq M\) and
Proof
Using the mean value theorem, we have
where ξ is a point between x and \(\frac{p^{n_{m}-k}[k]_{p,q}+\alpha }{[n]_{p,q}+\beta}b_{n}\). By using the above identity, we get
Hence,
since
Using it, we have
Now using Cauchy-Schwarz inequality for the second term, we obtain
Using Lemma 2, we see
Thus,
Choosing \(\delta:=(B_{n,p,q}(\alpha,\beta))^{1/2} \), we get the desired result. □
3.3 Voronovskaja-type result
Now, we prove a Voronovskaja-type approximation theorem with the help of the \(C_{n,m}^{(\alpha,\beta)}\) family of linear operators defined by (2.1).
Lemma 4
Let \((p_{n})\) and \((q_{n})\) be two sequences satisfying (3.2) and \(x \in[0, E]\) where \(E \in\mathbb {R}^{+}\). Then we get
and
where \(a \in(0, 1)\).
Proof
We shall prove only (3.10) because the proof of (3.9) is similar. Let \(x \in[0, E]\). Then, by Lemma (2), we obtain, for all \(n \in\mathbb {N}\),
Now by taking the limit as \(n \to\infty\) in (3.11), we obtain
which completes the proof. □
In a similar way to Lemma 4 one can deduce the following lemma.
Lemma 5
Let \((p_{n})\) and \((q_{n})\) be two sequences satisfying (3.2) and \(x \in[0, E]\) where \(E \in\mathbb {R}^{+}\). There is a positive constants \(M_{0}(x)\) depending only on x such that
Theorem 7
Let \((p_{n})\) and \((q_{n})\) be two sequences with the property (3.2). For every \(f\in C_{1+x^{2}}^{0}[0, \infty)\) such that \(f',f'' \in C_{1+x^{2}}^{0}[0, \infty)\), then
uniformly in \(x \in[0, E]\).
Proof
Using the Taylor formula for \(f\in C_{1+x^{2}}^{0}\), we have
where the function \(\eta_{x}(\cdot)\) is the remainder, \(\lim_{t\to x}\eta_{x}(t)=0\). Since the operator \(C_{n,m}^{(\alpha,\beta)}\) is linear
for each \(n \in\mathbb {N}\). We will now show that
After application of the Cauchy-Schwarz inequality for the third term on the right hand side of (3.13), we find that
Let us take \(\eta^{2}_{x}(t)=\theta_{x}(t)\), \(x \geq0\), we obtain
We have \(f \in C_{1+x^{2}}^{0}\) i.e. \(\lim_{|x| \to\infty} \frac{|f(x)|}{1+x^{2}} =\) finite value, which means f is function with maximum order of x is 2. Henceforth x is of order 1 and 0, respectively, in \(f{'}\) and \(f{''}\), i.e. \(f{''}\) is constant.
We will get a finite value of the above limit because numerator is a polynomial in x having terms of degree less than or equal to four and \(f,f',f'' \in C_{1+x^{2}}^{0}\). Thus \(\theta_{x}(t) \in C_{1+x^{2}}^{0}\).
Moreover, \(\lim_{t\to x}\theta_{x}(t)=0\). From Theorem 2, we observe that
uniformly in \(x \in[0, E]\). One obtains from Lemma 5 that
From these last two relations, the inclusion (3.14) holds true. Now by taking the limit as \(n \to\infty\) in (3.13) and using Lemma (4), we conclude that
uniformly in \(x \in[0, E]\), which leads us to the desired assertion of Theorem 7. □
3.4 Example
With the help of Maple, we show a comparison of the \((p,q)\) Bernstein-stancu operator and the operator (2.1) to the function \(f(x) = \sin(x)\) under the following parameters: \(\alpha= 1\), \(\beta= 1\), \(p = 0.9\), \(q = 0.8\), \(n = 1\) and \(b_{n}=\ln(1+n)\) within the interval \([0,b_{1}]\) i.e. \([0,\log_{e} 11]\). We have found it to be convenient to investigate our series only for finite sums. More powerful equipments with higher speed can easily compute the more complicated infinite series in a similar manner.
It is clear from the Figure 1 that approximation by the operator (2.1) is better than by \((p,q)\) Bernstein-stancu operator for \(f(x)=\sin x\) and it can be improved further by taking appropriate values of m and sequence \(b_{n}\).
4 Conclusion
A better approximation of complex functions over the required interval \([0,b_{n}]\) can be attained using the Chlodowsky variant of the \((p,q)\) Bernstein-Stancu-Schurer operator for choosing suitable values of the sequence \(b_{n}\) and n compared to classical operators over the fixed interval \([0,1]\).
References
Bernstein, SN: Démostration du théoréme de Weierstrass fondée sur le calcul de probabilités. Commun. Soc. Math. Kharkow (2) 13, 1-2 (1912/1913)
Lupaş, A: A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, vol. 9, pp. 85-92 (1987)
Phillips, GM: On generalized Bernstein polynomials. In: Numerical Analysis, pp. 263-269. World Scientific, River Edge (1996)
Ibikli, E: Approximation by Bernstein-Chlodowsky polynomials. Hacet. J. Math. Stat. 32, 1-5 (2003)
Karsli, H, Gupta, V: Some approximation properties of q-Chlodowsky operators. Appl. Math. Comput. 195, 220-229 (2008)
Mursaleen, M, Ansari, KJ, Khan, A: On \((p,q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874-882 (2015). Erratum: Appl. Math. Comput. 278, 70-71 (2016)
Mursaleen, M, Ansari, KJ, Khan, A: Some approximation results by \((p,q)\)-analogue of Bernstein-Stancu operators. Appl. Math. Comput. 264, 392-402 (2015). Corrigendum: Appl. Math. Comput. 269, 744-746 (2015)
Mursaleen, M, Nasiruzzaman, M, Nurgali, A: Some approximation results on Bernstein-Schurer operators defined by \((p,q)\)-integers. J. Inequal. Appl. 2015, Article ID 249 (2015). doi:10.1186/s13660-015-0767-4
Mursaleen, M, Alotaibi, A, Ansari, KJ: On a Kantorovich variant of \((p,q)\)-Szász-Mirakjan operators. J. Funct. Spaces 2016, Article ID 1035253 (2016)
Acar, T: \((p, q)\)-Generalization of Szász-Mirakyan operators. Math. Methods Appl. Sci. 39(10), 2685-2695 (2016)
Cai, QB, Zhou, G: On \((p,q)\)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators. Appl. Math. Comput. 276, 12-20 (2016)
Içöz, G, Mohapatra, RN: Weighted approximation properties of Stancu type modification of q-Szász-Durrmeyer operators. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 65(1), 87-103 (2016)
Içöz, G, Mohapatra, RN: Approximation properties by q-Durrmeyer-Stancu operators. Anal. Theory Appl. 29(4), 373-383 (2013)
Mishra, VN, Khatri, K, Mishra, LN, Deepmala: Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators. J. Inequal. Appl. 2013, Article ID 586 (2013)
Mishra, VN, Pandey, S: On \((p, q)\) Baskakov-Durrmeyer-Stancu operators. Adv. Appl. Clifford Algebras (2016). doi:10.1007/s00006-016-0738-y
Mishra, VN, Pandey, S: On Chlodowsky variant of \((p, q)\) Kantorovich-Stancu-Schurer operators. Int. J. Anal. Appl. 11(1), 28-39 (2016)
Mursaleen, M, Khan, F, Khan, A: Approximation properties for modified q-Bernstein-Kantorovich operators. Numer. Funct. Anal. Optim. 36(9), 1178-1197 (2015)
Mursaleen, M, Ansari, KJ, Khan, A: Stability of some positive linear operators on compact disk. Acta Math. Sci. 35B(6), 1-9 (2015)
Mursaleen, M, Khan, F, Khan, A: Approximation properties for King’s type modified q-Bernstein–Kantorovich operators. Math. Methods Appl. Sci. 2015(38), 5242-5252 (2015)
Sahai, V, Yadav, S: Representations of two parameter quantum algebras and \((p, q)\)-special functions. J. Math. Anal. Appl. 335, 268-279 (2007)
Ibikli, E: On Stancu type generalization of Bernstein-Chlodowsky polynomials. Mathematica 42(65), 37-43 (2000)
Gadjiev, AD: The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of P.P. Korovkin. Dokl. Akad. Nauk SSSR 218(5), 1001-1004 (1974). English translation in Sov. Math. Dokl. 15(5), 1433-1436
Vedi, T, Özarslan, MA: Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. J. Inequal. Appl. 2014, Article ID 189 (2014). doi:10.1186/1029-242X-2014-189
Acknowledgements
The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research article. The third author, SP, acknowledges MHRD, New Delhi, India for supporting this research article to carry out her research work (Ph.D.) under the supervision of Dr. Vishnu Narayan Mishra at Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev, Dumas Road, Surat (Gujarat), India, under FIR category.
Author information
Authors and Affiliations
Corresponding author
Additional information
Funding
The fourth author, AA, gratefully acknowledges the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Mishra, V.N., Mursaleen, M., Pandey, S. et al. Approximation properties of Chlodowsky variant of \((p,q)\) Bernstein-Stancu-Schurer operators. J Inequal Appl 2017, 176 (2017). https://doi.org/10.1186/s13660-017-1451-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-017-1451-7