Abstract
This paper is concerned with the \((p,q)\)-analog of Bernstein operators. It is proved that, when the function is convex, the \((p,q)\)-Bernstein operators are monotonic decreasing, as in the classical case. Also, some numerical examples based on Maple algorithms that verify these properties are considered. A global approximation theorem by means of the Ditzian-Totik modulus of smoothness and a Voronovskaja type theorem are proved.
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1 Introduction and preliminaries
During the last decade, the applications of q-calculus in the field of approximation theory has led to the discovery of new generalizations of classical operators. Lupaş [1] was first to observe the possibility of using q-calculus in this context. For more comprehensive details the reader should consult monograph of Aral et al. [2] and the recent references [3–9].
Nowadays, the generalizations of several operators in post-quantum calculus, namely the \((p,q)\)-calculus have been studied intensively. The \((p,q)\)-calculus has been used in many areas of sciences, such as oscillator algebra, Lie group theory, field theory, differential equations, hypergeometric series, physical sciences (see [10, 11]). Recently, Mursaleen et al. [12] defined \((p,q)\)-analog of Bernstein operators. The approximation properties for these operators based on Korovkin’s theorem and some direct theorems were considered. Also, many well-known approximation operators have been introduced using these techniques, such as Bleimann-Butzer-Hahn operators [13] and Szász-Mirakyan operators [14].
In the present paper, we prove new approximation properties of \((p,q)\)-analog of Bernstein operators. First of all, we recall some notations and definitions from the \((p,q)\)-calculus. Let \(0< q< p\leq1\). For each non-negative integer \(n\geq k\geq0\), the \((p,q)\)-integer \([k]_{p,q}\), \((p,q)\)-factorial \([k]_{p,q}!\), and \((p,q)\)-binomial are defined by
and
As a special case when \(p=1\), the above notations reduce to q-analogs.
The \((p,q)\)-power basis is defined as
The \((p,q)\)-derivative of the function f is defined as
Let f be an arbitrary function and \(a\in\mathbb{R}\). The \((p,q)\)-integral of f on \([0,a]\) is defined as
The \((p,q)\)-analog of Bernstein operators for \(x\in[0,1]\) and \(0< q< p\leq1\) are introduced as follows:
where the \((p,q)\)-Bernstein basis is defined as
Lemma 1.1
For \(x\in[0,1]\), \(0< q< p\leq1\), we have
where \(e_{i}(x)=x^{i}\) and \(i\in\{0,1,2\}\).
Lemma 1.2
Let n be a given natural number, then
where \(\phi(x)=\sqrt{x(1-x)}\) and \(x\in[0,1]\).
2 Monotonicity for convex functions
Oru and Phillips [15] proved that when the function f is convex on \([0,1]\), its q-Bernstein operators are monotonic decreasing. In this section we will study the monotonicity of \((p,q)\)-Bernstein operators.
Theorem 2.1
If f is convex function on \([0,1]\), then
for all \(n\geq1\) and \(0< q< p\leq1\).
Proof
We consider the knots \(x_{k}= \frac{p^{n-k}[k]_{p,q}}{[n]_{p,q}}\), \(\lambda_{k}=\bigl [ \scriptsize{\begin{array}{@{}c@{}} n\\ k \end{array}} \bigl ]_{p,q}p^{[k(k-1)-n(n-1)]/2}x^{k}(1\ominus x)_{p,q}^{n-k}\), \(0\leq k\leq n\).
Using Lemma 1.1, it follows that
From the convexity of the function f, we get
□
Example 2.2
Let \(f:\mathbb{R}\to\mathbb{R}\), \(f(x)=xe^{x+1}\). Figure 1 illustrates that \(B_{n}^{p,q}(f;x)\geq f(x)\) for the convex function f and \(x\in[0,1]\).
Theorem 2.3
Let f be convex on \([0,1]\). Then \(B_{n-1}^{p,q}(f;x)\geq B_{n}^{p,q}(f;x)\) for \(0< q< p\leq1\), \(0\leq x\leq 1\), and \(n\geq 2\). If \(f\in C[0,1]\) the inequality holds strictly for \(0< x<1\) unless f is linear in each of the intervals between consecutive knots \(\frac{p^{n-1-k}[k]_{p,q}}{[n-1]_{p,q}}\), \(0\leq k\leq n-1\), in which case we have the equality.
Proof
For \(0< q< p\leq1\) we begin by writing
Denote
and using the following relation:
we find
where
From (2.1) it is clear that each \(\Psi_{k}(x)\) is non-negative on \([0,1]\) for \(0< q< p\leq1\) and, thus, it suffices to show that each \(a_{k}\) is non-negative.
Since f is convex on \([0,1]\), then for any \(t_{0},t_{1}\in[0,1]\) and \(\lambda\in[0,1]\), it follows that
If we choose \(t_{0}=\frac{p^{n-k}[k-1]_{p,q}}{[n-1]_{p,q}}\), \(t_{1}=\frac{p^{n-1-k}[k]_{p,q}}{[n-1]_{p,q}}\), and \(\lambda=\frac{[k]_{p,q}}{[n]_{p,q}}q^{n-k}\), then \(t_{0},t_{1}\in[0,1]\) and \(\lambda\in(0,1)\) for \(1\leq k\leq n-1\), and we deduce that
Thus \(B_{n-1}^{p,q}(f;x)\geq B_{n}^{p,q}(f;x)\).
We have equality for \(x=0\) and \(x=1\), since the Bernstein polynomials interpolate f on these end-points. The inequality will be strict for \(0< x<1\) unless when f is linear in each of the intervals between consecutive knots \(\frac{p^{n-1-k}[k]_{p,q}}{[n-1]_{p,q}}\), \(0\leq k\leq n-1\), then we have \(B_{n-1}^{p,q}(f;x)=B_{n}^{p,q}(f;x)\) for \(0\leq x\leq1\). □
Example 2.4
Let \(f(x)=\sin(2\pi x)\), \(x\in[0,1]\). Figure 2 illustrates the monotonicity of \((p,q)\)-Bernstein operators for \(p=0.95\) and \(q=0.9\). We note that if f is increasing (decreasing) on \([0,1]\), then the operators is also increasing (decreasing) on \([0,1]\).
3 A global approximation theorem
In the following we establish a global approximation theorem by means of Ditzian-Totik modulus of smoothness. In order to prove our next result, we recall the definitions of the Ditzian-Totik first order modulus of smoothness and the K-functional [16]. Let \(\phi(x) =\sqrt{x(1-x)}\) and \(f\in C[0,1]\). The first order modulus of smoothness is given by
The corresponding K-functional to (3.1) is defined by
where \(W_{\phi}[0,1]=\{g:g\in AC_{\mathrm{loc}}[0,1],\|\phi g^{\prime }\|<\infty\}\) and \(g\in AC_{\mathrm{loc}}[0,1]\) means that g is absolutely continuous on every interval \([a,b]\subset[0,1]\). It is well known ([16], p.11) that there exists a constant \(C>0\) such that
Theorem 3.1
Let \(f\in C[0,1] \) and \(\phi(x) =\sqrt{x(1-x)}\), then for every \(x\in[0,1]\), we have
where C is a constant independent of n and x.
Proof
Using the representation
we get
For any \(x\in(0,1)\) and \(t\in[0,1]\) we find that
Further,
From (3.3)-(3.5) and using the Cauchy-Schwarz inequality, we obtain
Using Lemma 1.2, we get
Now, using the above inequality we can write
Taking the infimum on the right-hand side of the above inequality over all \(g\in W_{\phi}[0,1]\), we get
Using equation (3.2) this theorem is proven. □
4 Voronovskaja type theorem
Using the first order Ditzian-Totik modulus of smoothness, we prove a quantitative Voronovskaja type theorem for the \((p,q)\)-Bernstein operators.
Theorem 4.1
For any \(f\in C^{2}[0,1]\) the following inequalities hold:
-
(i)
\(|[n]_{p,q} [B_{n}^{p,q}(f;x)-f(x) ]-\frac {p^{n-1}\phi^{2}(x)}{2}f^{\prime\prime}(x) |\leq C\omega_{\phi} (f^{\prime\prime},\phi(x)n^{-1/2} )\),
-
(ii)
\(|[n]_{p,q} [B_{n}^{p,q}(f;x)-f(x) ]-\frac {p^{n-1}\phi^{2}(x)}{2}f^{\prime\prime}(x) |\leq C\phi(x)\omega_{\phi} (f^{\prime\prime},n^{-1/2} )\),
where C is a positive constant.
Proof
Let \(f\in C^{2}[0,1]\) be given and \(t,x\in[0,1]\). Using Taylor’s expansion, we have
Therefore,
In view of Lemma 1.1 and Lemma 1.2, we get
The quantity \(\vert \int_{x}^{t}\vert f^{\prime\prime}(u)-f^{\prime \prime}(x)\vert |t-u|\,du\vert \) was estimated in [17], p.337, as follows:
where \(g\in W_{\phi}[0,1]\). On the other hand, for any \(m=1,2,\ldots\) and \(0< q< p\leq1\), there exists a constant \(C_{m}>0\) such that
where \(x\in[0,1]\) and \(\lfloor a\rfloor\) is the integer part of \(a\geq0\).
Throughout this proof, C denotes a constant not necessarily the same at each occurrence.
Now, combining (4.1)-(4.3) and applying Lemma 1.2, the Cauchy-Schwarz inequality, we get
Since \(\phi^{2}(x)\leq\phi(x)\leq1\), \(x\in[0,1]\), we obtain
Also, the following inequality can be obtained:
Taking the infimum on the right-hand side of the above relations over \(g\in W_{\phi}[0,1]\), we get
5 Better approximation
In 2003, King [18] proposed a technique to obtain a better approximation for the well-known Bernstein operators as follows:
where \(r_{n}\) is a sequence of continuous functions defined on \([0,1]\) with \(0\leq r_{n}(x)\leq1\) for each \(x\in[0,1]\) and \(n\in\{ 1,2,\ldots\}\). The modified Bernstein operators (5.1) preserve \(e_{0}\) and \(e_{2}\) and present a degree of approximation at least as good. In [19], the authors consider the sequence of linear Bernstein-type operators defined for \(f\in C[0,1]\) by \({B}_{n}(f\circ\tau^{-1})\circ\tau\), τ being any function that is continuously differentiable ∞ times on \([0,1]\), such that \(\tau(0)=0\), \(\tau(1)=1\), and \(\tau^{\prime}(x)>0\) for \(x\in[0,1]\).
So, using the technique proposed in [19], we modify the \((p,q)\)-Bernstein operators as follows:
where
Then we have
where \(\phi_{\tau}^{2}(x):=\tau(x)(1-\tau(x))\).
Example 5.1
We compare the convergence of \((p,q)\)-analog of Bernstein operators \(B_{n}^{p,q}f\) with the modified operators \(\overline{B}_{n}^{p,q}f\). We have considered the function \(f(x)=\sin(10x)\) and \(\tau(x)= \frac{x^{2}+x}{2}\). For \(x\in [\frac{1}{2},1 ]\), \(p=0.95\), \(q=0.9\), \(n=100\), the convergence of the operators \(B_{n}^{p,q}\) and \(\overline{B}_{n}^{p,q}\) to the function f is illustrated in Figure 3. Note that the approximation by \(\overline{B}_{n}^{p,q}f\) is better than using \((p,q)\)-Bernstein operators \(B_{n}^{p,q}f\).
References
Lupaş, A: A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus, vol. 9, pp. 85-92. University of Cluj-Napoca, Cluj-Napoca (1987)
Aral, A, Gupta, V, Agarwal, RP: Applications of q-Calculus in Operator Theory. Springer, New York (2013)
Acu, AM: Stancu-Schurer-Kantorovich operators based on q-integers. Appl. Math. Comput. 259, 896-907 (2015). doi:10.1016/j.amc.2015.03.032
Acar, T, Aral, A: On pointwise convergence of q-Bernstein operators and their q-derivatives. Numer. Funct. Anal. Optim. 36(3), 287-304 (2015). doi:10.1080/01630563.2014.970646
Acu, AM, Muraru, CV: Approximation properties of bivariate extension of q-Bernstein-Schurer-Kantorovich operators. Results Math. 67(3-4), 265-279 (2015). doi:10.1007/s00025-015-0441-7
Agratini, O: On a q-analogue of Stancu operators. Cent. Eur. J. Math. 8(1), 191-198 (2010). doi:10.2478/s11533-009-0057-9
Kang, SM, Acu, AM, Rafiq, A, Kwun, YC: Approximation properties of q-Kantorovich-Stancu operator. J. Inequal. Appl. 2015, 211 (2015). doi:10.1186/s13660-015-0729-x
Kang, SM, Acu, AM, Rafiq, A, Kwun, YC: On q-analogue of Stancu-Schurer-Kantorovich operators based on q-Riemann integral. J. Comput. Anal. Appl. 21(3), 564-577 (2016)
Ulusoy, G, Acar, T: q-Voronovskaya type theorems for q-Baskakov operators. Math. Methods Appl. Sci. (2015). doi:10.1002/mma.3784
Burban, I: Two-parameter deformation of the oscillator algebra and \((p,q)\)-analog of two-dimensional conformal field theory. J. Nonlinear Math. Phys. 2(3-4), 384-391 (1995). doi:10.2991/jnmp.1995.2.3-4.18
Sahai, V, Yadav, S: Representations of two parameter quantum algebras and \(p,q\)-special functions. J. Math. Anal. Appl. 335, 268-279 (2007). doi:10.1016/j.jmaa.2007.01.072
Mursaleen, M, Ansari, KJ, Khan, A: Erratum to ‘On \((p, q)\)-analogue of Bernstein operators’ [Appl. Math. Comput. 266 (2015) 874-882]. Appl. Math. Comput. 278, 70-71 (2016). doi:10.1016/j/amc.2015.04.090
Mursaleen, M, Nasiruzzaman, M, Khan, A, Ansari, KJ: Some approximation results on Bleimann-Butzer-Hahn operators defined by \((p, q)\)-integers. Filomat 30(3), 639-648 (2016). doi:10.2298/FIL1603639M
Acar, T: \((p,q)\)-Generalization of Szász-Mirakyan operators. Math. Methods Appl. Sci. (2015). doi:10.1002/mma.3721
Oru, H, Phillips, GM: A generalization of the Bernstein polynomials. Proc. Edinb. Math. Soc. 42, 403-413 (1999). doi:10.1017/S0013091500020332
Ditzian, Z, Totik, V: Moduli of Smoothness. Springer, New York (1987)
Finta, Z: Remark on Voronovskaja theorem for q-Bernstein operators. Stud. Univ. Babeş-Bolyai, Math. 56(2), 335-339 (2011)
King, JP: Positive linear operators which preserve \(x^{2}\). Acta Math. Hung. 99, 203-208 (2003). doi:10.1023/A:1024571126455
Cárdenas-Morales, D, Garrancho, P, Raşa, I: Bernstein-type operators which preserve polynomials. Comput. Math. Appl. 62(1), 158-163 (2011). doi:10.1016/j.camwa.2011.04.063
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The authors would like to thank the editor and the referees for useful comments and suggestions. This work was supported by the Dong-A University research fund.
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Kang, S.M., Rafiq, A., Acu, AM. et al. Some approximation properties of \((p,q)\)-Bernstein operators. J Inequal Appl 2016, 169 (2016). https://doi.org/10.1186/s13660-016-1111-3
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DOI: https://doi.org/10.1186/s13660-016-1111-3