Abstract
Our purpose is to introduce a two-parametric \((p, q)\)-analogue of the Stancu-Beta operators. We study approximating properties of these operators using the Korovkin approximation theorem and also study a direct theorem. We also obtain the Voronovskaya-type estimate for these operators. Furthermore, we study the weighted approximation results and pointwise estimates for these operators.
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1 Introduction
The q-calculus has attracted attention of many researchers because of its applications in various fields such as numerical analysis, computer-aided geometric design, differential equations, and so on. In the field of approximation theory, the application of q-calculus has been the area of many recent researches.
Lupaş [1] presented the first q-analogue of the classical Bernstein operators in 1987. He studied the approximation and shape-preserving properties of these operators. Another q-companion of the classical Bernstein polynomials is due to Phillips [2]. Inspired by this, several authors produced generalizations of well-known positive linear operators based on q-integers and studied them extensively. For instance, the approximation properties of the Kantorovich-type q-Bernstein operators [3], q-BBH operators [4], q-analogue of generalized Bernstein-Schurer operators [5], weighted statistical approximation by Kantorovich-type q-Szász-Mirakjan operators [6], q-Szász-Durrmeyer operators [7], operators constructed by means of q-Lagrange polynomials and A-statistical approximation [8], statistical approximation properties of modified q-Stancu-Beta operators [9], and q-Bernstein-Schurer-Kantorovich operators [10].
The q-calculus has led to the discovery of the \((p,q)\)-calculus. Recently, Mursaleen et al. have used the \((p,q)\)-calculus in approximation theory. They have applied it to construct a \((p,q)\)-analogue of the classical Bernstein operators [11], a \((p,q)\)-analogue of the Bernstein-Stancu operators [12], and a \((p,q)\)-analogue of the Bernstein-Schurer operators [13] and have studied their approximation properties. Most recently, \((p,q)\)-analogues of some other operators have been studied in [14–18], and [19].
We now give some basic notions of the \((p,q)\)-calculus.
The \((p,q)\)-integer is defined by
The \((p,q)\)-companion of the binomial expansion is
The \((p,q)\)-analogues of the binomial coefficients are defined by
The \((p,q)\)-analogues of definite integrals of a function f are defined by
and
For \(m,n\in N\), the \((p,q)\)-gamma and the \((p,q)\)-beta functions are defined by
and
respectively. These two are related by
For \(p=1\), all the concepts of the \((p,q)\)-calculus reduce to those of q-calculus. The details on \((p,q)\)-calculus can be found in [20–22].
Stancu [23] introduced the beta operators to approximate the Lebesgue-integrable functions on \([0,\infty)\) as follows:
The q-companion of the Stancu-Beta operators was given by Aral and Gupta [24] as follows:
Let \(0< q< p<1\). Mursaleen et al. [25] constructed the \((p,q)\)-Stancu-Beta operators as follows:
They investigated the approximating properties and estimated the rate of convergence of these operators. Motivated by this work, we introduce the following sequence of operators:
where \(0\leq\alpha\leq\beta\). We call them two-parametric \((p,q)\)-Stancu-Beta operators. For \(\alpha=0=\beta\), the operators (1.4) coincide with the operators (1.3). So the latter is a generalization of the former.
2 Main results
We shall investigate approximation results for the operators (1.4). We calculate the moments of the operators \(S_{n,p,q}^{\alpha,\beta }(f;x)\) in the following lemma.
Lemma 2.1
Let \(S_{n,p,q}^{\alpha,\beta}(f;x)\) be given by (1.4). Then we have the following equalities:
-
(i)
\(S_{n,p,q}^{\alpha,\beta}(1;x)=1\),
-
(ii)
\(S_{n,p,q}^{\alpha,\beta}(t;x)=\frac{[n]_{p,q}}{([n]_{p,q}+\beta)}x+\frac{\alpha}{([n]_{p,q}+\beta)}\),
-
(iii)
\(S_{n,p,q}^{\alpha,\beta}(t^{2};x)=\frac{[n]_{p,q}^{3}}{pq([n]_{p,q}-1)([n]_{p,q}+\beta)^{2}}x^{2}+\frac{[n]_{p,q}}{([n]_{p,q}+\beta)^{2}} (\frac{[n]_{p,q}}{pq([n]_{p,q}-1)}+2\alpha )x+\frac{\alpha^{2}}{([n]_{p,q}+\beta)^{2}}\).
Proof
Using (1.1), (i) is immediate. Further,
and (ii) is proved;
which proves (iii).
Hence, the lemma is proved. □
We readily obtain the following lemma.
Lemma 2.2
Let \(p,q\in(0,1)\). Then, for \(x\in[0,\infty)\), we have:
-
(i)
\(S_{n,p,q}^{\alpha,\beta}((t-x);x)=\frac{\alpha-\beta x}{([n]_{p,q}+\beta)}\),
-
(ii)
\(S_{n,p,q}^{\alpha,\beta}((t-x)^{2};x)\leq (\frac {[n]_{p,q}}{pq([n]_{p,q}-1)}-\frac{([n]_{p,q}-\beta)}{([n]_{p,q}+\beta)} )x^{2}+\frac{1}{pq([n]_{p,q}-1)}x+\frac{\alpha^{2}}{([n]_{p,q}+\beta )^{2}}\leq \frac{2(1+\beta)^{2}x^{2}+x+\alpha^{2}}{pq([n]_{p,q}-1)}\).
Proof
We have
which proves (i). Now
which gives (ii). Hence, the lemma is proved. □
Next, we present a direct theorem for the operators \(S_{n,p,q}^{\alpha ,\beta}(f;x)\).
We denote By \(C_{B}[0,\infty)\), the space of all real-valued continuous bounded functions f on the interval \([0,\infty)\) endowed with the norm
Let \(\delta>0\) and \(W^{2}= \{ h:h^{\prime},h^{\prime\prime}\in C(I), I= [0, \infty) \} \), then the Peetre K-functional is defined by
The second-order modulus of continuity \(\omega_{2}\) of f is defined as
By DeVore-Lorentz theorem (see [26], p.177, Theorem 2.4) there exists a constant \(C>0\) such that
Also, by \(\omega(f,\delta)\) we denote the first-order modulus of continuity of \(f\in C(I)\) defined as
We shall use the notation \(v^{2}(x)=x+x^{2}\).
Theorem 2.3
Suppose that \(f\in C_{B}[0,\infty)\) and \(0< p,q<1\). Then for all \(x\in[0,\infty)\) and \(n\geq2\), there exists a constant C such that
where
and
Proof
Let us define the auxiliary operators
By the Lemma 2.1 it is readily seen that these operators are linear and
Suppose that \(g\in W^{2}\). By the Taylor expansion we can write
Operating by \(S_{n,p,q}^{\ast\alpha,\beta}(.;x)\) on both sides of the above and using (2.3), we obtain:
Using (2.2) in the right-hand side, we get
So we obtain
Using the linearity of the integral operator and the operator \(S_{n,p,q}^{\alpha ,\beta}(\cdot;x)\) in the second and first parts of right-hand side, respectively, and using the fact that for all \(x \in[0, \infty)\),
we obtain
In the first part, solving the integral \(\int_{x}^{t} | t-u|\,du\) and using the linearity of the operators \(S_{n,p,q}^{\alpha,\beta}(\cdot;x)\), we readily see that
and after some calculations, for the second part of (2.4), we get
So by (2.4), we obtain
Using Lemma 2.2(ii), we obtain
where
Therefore, by (2.5) we get
On the other hand, by (2.2) we have
By (2.2), (2.6), and (2.7), we obtain:
where
Taking the infimum over all \(g\in W^{2}\) on the right-hand side of (2.8), we obtain
Using relation (2.1), for \(p,q\in(0,1)\), we get
and this completes the proof. □
3 Rate of approximation
Let \(B_{x^{2}}[0,\infty)\) denote the set of all functions f such that \(f(x)\leq M_{f}(1+x^{2})\), where \(M_{f}\) is a constant depending on f. By \(C_{x^{2}}[0,\infty)\) we denote the subspace of all continuous functions in the space \(B_{x^{2}}[0,\infty)\). Also, we denote by \(C_{x^{2}}^{\ast }[0,\infty)\), the subspace of all functions \(f\in C_{x^{2}}[0,\infty )\) for which \(\lim_{x\rightarrow\infty}\frac{f(x)}{1+x^{2}}\) is finite with
For \(a>0\), the modulus of continuity of f over \([0,a]\) is defined by
We have the following proposition.
Proposition 3.1
-
(i)
For \(f\in C_{x^{2}}[0,\infty)\), the modulus of continuity \(\omega _{a}(f,\delta)\), \(a>0\), approaches to zero.
-
(ii)
For every \(\delta>0\), we have
$$ \bigl|f(y)-f(x)\bigr| \leq \biggl(1+\frac{|y-x|}{\delta} \biggr)\omega_{a}(f, \delta) $$and
$$ \bigl|f(y)-f(x)\bigr| \leq \biggl(1+\frac{(y-x)^{2}}{\delta^{2}} \biggr)\omega _{a}(f, \delta). $$
In the following theorem, we estimate the rate of convergence of the operators \(S_{n,p,q}^{\alpha,\beta}(f;x)\).
Theorem 3.2
Let \(f\in C_{x^{2}}[0,\infty),p,q\in(0,1)\), and let \(\omega _{a+1}(f,\delta)\) be the modulus of continuity on the interval \([0,1+a]\subseteq[0,\infty),a>0\). Then, for \(n\geq2\), we have
Proof
Let \(x\in[0,a]\) and \(t> a+1\). Since \(1+x< t\), we have
For \(\delta>0\), \(x\in[0,a]\), \(t-1\leq a\), by Proposition 3.1 we obtain
By (3.1) and (3.2), for \(x\in[0,a]\) and nonnegative t, we can write
Therefore,
Hence, using the Lemma 2.2(ii) and the Schwarz inequality, for every \(p,q\in(0,1)\) and \(x\in[0,a]\), we obtain
By choosing \(\delta^{2}=\frac{2(1+\beta)^{2}a^{2}+a+\alpha^{2}}{pq([n]_{p,q}-1)}\) we get the required result. □
4 Weighted approximation
This section is devoted to the study of weighted approximation theorems for the operators (2.2).
Theorem 4.1
Suppose that \(p=p_{n}\) and \(q=q_{n}\) are two sequences satisfying \(0< p_{n},q_{n}<1\) and suppose that \(p_{n}\rightarrow1\) and \(q_{n}\rightarrow1\) as \(n\rightarrow \infty\). Then, for each \(f\in C_{x^{2}}^{\ast}[0,\infty)\), we have
Proof
By the theorem in [27] it suffices to prove that
By Lemma 2.1(i)-(ii), the conditions of (4.1) are easily verified for \(i=0\) and 1. For \(i=2\), we can write
which implies that
This completes the proof of the theorem. □
Theorem 4.2
Let \(p=(p_{n})\) and \(q=(q_{n})\) be two sequences such that \(0< p_{n}, q_{n}<1\), and let \(p_{n}\rightarrow1\) and \(q_{n}\rightarrow1\) as \(n\rightarrow \infty\). Then, for each \(f\in C_{x^{2}}[0,\infty)\) and all \(\alpha>0\), we have
Proof
For \(x_{0}>0\) fixed, we have:
The first term of this inequality goes to zero by Theorem 3.2. Also, for any fixed \(x_{0}>0\), it is readily seen from Lemma 2.1 that
approaches zero as \(n\rightarrow\infty\). If we choose \(x_{0}>0\) large enough so that the last part of the last inequality is arbitrarily small, then our theorem is proved. □
5 Voronovskaya-type theorem
This section presents the Voronovskaya-type theorem for the operators \(S_{n,p,q}^{\alpha,\beta}(f;x)\). We need the following lemma.
Lemma 5.1
Suppose that \(p_{n},q_{n}\in(0,1)\) are such that \(p_{n}^{n}\rightarrow a,q_{n}^{n}\rightarrow b \) (\(0\leq a,b<1\)) as \(n\rightarrow\infty\). Then, for every \(x\in[0,\infty)\), simple computations yield
Theorem 5.2
Assume that \(p_{n},q_{n}\in(0,1)\) are such that \(p_{n}^{n}\rightarrow a,q_{n}^{n}\rightarrow b \) (\(0\leq a,b<1\)) as \(n\rightarrow\infty\). Then, for \(f\in C_{x^{2}}^{\ast}[0,\infty)\) such that \(f^{\prime },f_{x^{2}}^{\prime \prime\ast}[0,\infty)\), we have
uniformly on \([0,A]\) for any \(A>0\).
Proof
Let \(f, f', f''\in C^{*}_{x^{2}}[0,\infty)\) and \(x \in[0, \infty)\). By the Taylor formula we can write
where \(r(t;x)\) is the remainder term, \(r(\cdot;x) \in C^{*}_{x^{2}}[0,\infty)\), and \(\lim_{t\rightarrow x} r(t;x)=0\). Operating by \(S_{n,p_{n},q_{n}}^{\alpha,\beta}\) on both sides of (5.1), we get
It follows from the Cauchy-Schwarz inequality that
Note that \(r^{2}(x;x)=0\) and \(r^{2}(\cdot;x)\in C^{*}_{x^{2}}[0,\infty)\). Therefore, it follows that
uniformly over \([0, A]\).
By Lemma 5.1 and equations (5.2) and (5.3), we obtain
Thus, we obtain
□
6 Pointwise estimates
In this section, we study pointwise estimates of rate of convergence of the operators \(S_{n,p,q}^{\alpha,\beta}(f;x)\).
Let \(0<\nu\leq\) and \(E\subset[0,\infty)\). We say that a function \(f\in C[0,\infty)\) belongs to \(Lip(\nu)\) if
where \(M_{f}\) is a constant depending on α and f only.
We have the following theorem.
Theorem 6.1
Let \(\nu\in(0,1],f\in Lip(\nu)\), and \(E\subset[0,\infty)\). Then, for \(x\in[0,\infty)\),
where \(d(x,E)\) denotes the distance of the point x from the set E, defined by
Proof
Taking \(y\in\bar{E}\), we can write
By (6.1) we have
Using the Hölder inequality with \(p=\frac{2}{\nu},q=\frac{2}{2-\nu }\), we obtain
and the theorem is proved. □
We now present a theorem regarding a local direct estimate for the operators \(S_{n,p,q}^{\alpha,\beta}(f;x)\) in terms of the Lipschitz-type maximal function of order ν as introduced by Lenze [28]. It is defined by
Theorem 6.2
Let \(\nu\in(0,1]\) and \(f\in C[0,\infty)\). Then, for each \(x\in [ 0,\infty)\), we have
Proof
By (6.2) we can write
and
Using the Lemma 2.2 and applying the Hölder inequality with \(p=\frac {2}{\nu}\), \(q= \frac{2}{2-\nu}\), we obtain
which proves the theorem. □
Remark
The further properties of the operators such as convergence properties via summability methods (see, e.g., [29–31]) can be studied.
7 Conclusions
In this paper, we have introduced a two-parametric \((p,q)\)-analogue of the Stancu-Beta operators and studied some approximating properties of these operators. We also obtained the Voronovskaya-type estimate and the weighted approximation results for these operators. Furthermore, we obtained a pointwise estimate for these operators.
References
Lupaş, A: A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus, vol. 87, pp. 85-92. Univ. ‘Babeş-Bolyai’, Cluj-Napoca (1987)
Phillips, GM: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511-518 (1997)
Dalmanoglu, Ö: Approximation by Kantorovich type q-Bernstein operators. In: Proceedings of the 12th WSEAS International Conference on Applied Mathematics, Cairo, Egypt, pp. 113-117 (2007)
Aral, A, Doğru, O: Bleimann Butzer and Hahn operators based on q-integers. J. Inequal. Appl. 2007, Article ID 79410 (2007)
Muraru, CV: Note on q-Bernstein-Schurer operators. Stud. Univ. Babeş–Bolyai, Math. 56(2), 489-495 (2011)
Örkcü, M, Doğru, Ö: Weighted statistical approximation by Kantorovich type q-Szász-Mirakjan operators. Appl. Math. Comput. 217, 7913-7919 (2011)
Mahmudov, NI: On q-Szász-Durrmeyer operators. Cent. Eur. J. Math. 8(2), 399-409 (2010)
Mursaleen, M, Khan, A, Srivastava, HM, Nisar, KS: Operators constructed by means of q-Lagrange polynomials and A-statistical approximation. Appl. Math. Comput. 219, 6911-6918 (2013)
Mursaleen, M, Khan, A: Statistical approximation properties of modified q-Stancu-Beta operators. Bull. Malays. Math. Soc. 36(3), 683-690 (2013)
Özarslan, MA, Vedi, T: q-Bernstein-Schurer-Kantorovich Operators. J. Inequal. Appl. 2013, Article ID 444 (2013)
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, Nasiuzzaman, M, Nurgali, A: Some approximation results on Bernstein-Schurer operators defined by \((p,q)\)-integers. J. Inequal. Appl. 2015, Article ID 249 (2015)
Acar, T: \((p,q)\)-Generalization of Szasz-Mirakyan operators. Math. Methods Appl. Sci. 39(10), 2685-2695 (2016)
Acar, T, Aral, A, Mohiuddine, SA: On Kantorovich modifications of \((p,q)\)-Baskakov operators. J. Inequal. Appl. 2016, Article ID 98 (2016)
Acar, T, Aral, A, Mohiuddine, SA: Approximation by bivariate \((p, q)\)-Bernstein-Kantorovich operators. Iran. J. Sci. Technol., Trans. A, Sci. (2016). doi:10.1007/s40995-016-0045-4
Cai, QB, Zhou, G: On \((p,q)\)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators. Appl. Math. Comput. 276, 12-20 (2016)
Mursaleen, M, Alotaibi, A, Ansari, KJ: On a Kantorovich variant of \((p,q)\)-Szász-Mirakjan operators. J. Funct. Spaces 2016, Article ID 1035253 (2016)
Mursaleen, M, Khan, F, Khan, A: Approximation by \((p,q)\)-Lorentz polynomials on a compact disk. Complex Anal. Oper. Theory (2016). doi:10.1007/s11785-016-0553-4
Hounkonnou, MN, Dsir, J, Kyemba, B: \(R(p,q)\)-Calculus: differentiation and integration. SUT J. Math. 49(2), 145-167 (2013)
Sadjang, PN: On the fundamental theorem of \((p,q)\)-calculus and some \((p,q)\)-Taylor formulas. arXiv:1309.3934 [math.QA]
Sahai, V, Yadav, S: Representations of two parameter quantum algebras and \(p,q\)-special functions. J. Math. Anal. Appl. 335, 268-279 (2007)
Stancu, DD: On the beta approximating operators of second kind. Rev. Anal. Numér. Théor. Approx. 24, 231-239 (1995)
Aral, A, Gupta, V: On the q-analogue of Stancu-Beta operators. Appl. Math. Lett. 25, 67-71 (2012)
Mursaleen, M, Khan, T: Approximation by Stancu-Beta operators via \((p,q)\)-calculus. arXiv:1602.06319, submit/1463821
Devore, RA, Lorentz, GG: Constructive Approximation. Springer, Berlin (1993)
Gadjiev, AD: Theorems of the type of P.P. Korovkin type theorems. Mat. Zametki 20(5), 781-786 (1976) (English translation: Math. Notes 20(5-6), 996-998 (1976))
Lenze, B: On Lipschitz-type maximal functions and their smoothness spaces. Indag. Math. 91, 53-63 (1988)
Braha, NL, Srivastava, HM, Mohiuddine, SA: A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Pousin mean. Appl. Math. Comput. 228, 162-169 (2014)
Edely, OHH, Mohiuddine, SA, Noman, AK: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 23, 1382-1387 (2010)
Mohiuddine, SA: An application of almost convergence in approximation theorems. Appl. Math. Lett. 24, 1856-1860 (2011)
Acknowledgements
Research of the second author is supported by the Grants Nos. 5414/GF4, 0971/GF4 of the Committee of Science of Ministry of Education and Science of the Republic of Kazakhstan.
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Mursaleen, M., Sarsenbi, A.M. & Khan, T. On \((p,q)\)-analogue of two parametric Stancu-Beta operators. J Inequal Appl 2016, 190 (2016). https://doi.org/10.1186/s13660-016-1128-7
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DOI: https://doi.org/10.1186/s13660-016-1128-7