Abstract
Recently, a mixed hybrid operator, generalizing the well-known Phillips operators and Baskakov–Szász type operators, was introduced. In this paper, we study Bézier variant of these new operators. We investigate the degree of approximation of these operators by means of the Lipschitz class function, the modulus of continuity, and a weighted space. We study a direct approximation theorem by means of the unified Ditzian–Totik modulus of smoothness. Furthermore, the rate of convergence for functions having derivatives of bounded variation is discussed.
Similar content being viewed by others
1 Introduction
For a continuous function h on \([0,1]\), Bernstein [1] defined a linear positive operator in order to provide a very simple and elegant proof of the Weierstrass approximation theorem, namely
In order to approximate continuous functions on \([0,\infty)\), Szász [2] introduced the operator
provided the infinite series on the right-hand side converges. Later on, for \(h\in \operatorname {C}[0,\infty)\) and \(0\leq\beta<1\), Jain [3] proposed a modification of the operators given in (1.1), namely
where
with the partition of unity
It is observed that the Jain operator (1.2) includes the Szász operator (1.1) as a special case for \(\beta=0\). Recently, Gupta and Greubel [4] also proposed the Durrmeyer type modification of the operators given in (1.2) as
where
They showed that these operators converge to h without any restriction on β. The moments for these operators were obtained by using Tricomi’s hypergeometric functions and Stirling numbers of first kind, and some approximation properties of these operators were proved.
In the literature, many authors have discussed the approximation behavior of different summation-integral type operators (see [5, 6]). For \(0\leq\beta<1\) and \(c\geq0\), Acu and Gupta [7] introduced mixed Durrmeyer type operators for \(x\in[0, \infty)\) as
where
and
They determined the degree of approximation by means of the modulus of continuity and a weighted space. The authors also studied the approximation of functions having derivatives equivalent with a function of bounded variation. It is observed that the operator defined by (1.3) has two special cases:
-
(1)
If \(\phi_{n,c}(x)= e^{-nx}\) and \(\beta=0\), then the Phillips operators are obtained [8].
-
(2)
If \(\phi_{n,c}(x)=(1+cx)^{-n/c}\) and \(\beta=0\), then one gets the Baskakov–Szász operators [9].
Zeng and Piriou [10] initiated the study of Bézier variant of Bernstein operators. Zeng and Chen [11] estimated the rate of approximation for Bézier–Bernstein–Durrmeyer operators. Zeng and Tao [12] considered Bézier–Baskakov–Durrmeyer operators for \(\alpha\geq1\) and obtained the rate of convergence. For some other contributions in this direction, we refer to [13–21]. Motivated by the above research, we introduce the Bézier variant of the operator (1.3) as
where \(\alpha\geq1\),
and
δ being the Dirac delta function. For \(\alpha=1\), we note that \(P_{n,\alpha}^{\beta,c}(h;x)=P_{n}^{\beta,c}(h;x)\).
Recently, Acar et al. [22] considered the Bézier variant of Bernstein–Durrmeyer type operators and studied the degree of approximation of functions having derivative of bounded variation. The order of approximation of summation-integral type operators for functions with derivatives of bounded variation is estimated in [13, 23–27].
The aim of this paper is to investigate the weighted approximation properties and a direct approximation result by means of the Ditzian–Totik modulus of smoothness \(\omega_{\phi^{\tau}}(h;t)\), \(0\leq\tau\leq1\), and the rate of convergence for functions having a derivative of bounded variation for the operators given by (1.4). Throughout this paper, C denotes a constant which may be different at each occurrence.
2 Preliminaries
In the sequel, the following auxiliary results are used to prove the main results of the paper.
Lemma 1
(see [7])
For the mth order moment \(P_{n}^{\beta,c}(t^{m};x)\), \(m=0,1,2\), we obtain
-
(1)
\(P_{n}^{\beta,c}(1;x)=1\);
-
(2)
\(P_{n}^{\beta,c}(t;x)=(1-\beta)x +\frac{\beta(2-\beta)}{n(1- \beta)}(1-\phi_{n,c}(x))\);
-
(3)
\(P_{n}^{\beta,c}(t^{2};x)=(1-\beta)^{2} [ x^{2}+ \frac{x(1+cx)}{n} ] +\frac{(1+4\beta-2\beta^{2})}{n}x +\frac{ \beta^{2}(3-\beta)}{n^{2}(1-\beta)}(1-\phi_{n,c}(x))\).
Consequently, for the rth order central moment \(\mu_{n,r}^{\beta,c}(x)=P _{n}^{\beta,c}((t-x)^{r};x)\), \(r=0,1,2\), one has
-
(1)
\(\mu_{n,0}^{\beta,c}(x)=1\);
-
(2)
\(\mu_{n,1}^{\beta,c}(x) =-\beta x+\frac{\beta(2-\beta)}{n(1-\beta )}(1-\phi_{n,c}(x))\);
-
(3)
\(\mu_{n,2}^{\beta,c}(x)= [ \beta^{2}+\frac{c(1-\beta)^{2}}{n} ] x ^{2} +\frac{2-4\beta-\beta^{2}+\beta^{3}+2\beta(2-\beta)\phi _{n,c}(x)}{n(1- \beta)}x +\frac{\beta^{2}(3-\beta)(1-\phi_{n,c}(x))}{n^{2}(1-\beta)}\).
Lemma 2
(see [7])
If \(\beta=\beta(n)\to0\) as \(n\to\infty\) and \(\lim_{n\to\infty}n \beta(n)=l\in {\mathbb {R}}\), then
-
(1)
\(\lim_{n\to\infty}n\mu_{n,1}^{\beta,c}(x)=-lx\);
-
(2)
\(\lim_{n\to\infty}n\mu_{n,2}^{\beta,c}(x)=x(cx+2)\);
-
(3)
\(\lim_{n\to\infty}n^{2}\mu_{n,4}^{\beta,c}(x)=3x^{2}(cx+2)^{2}\).
Remark 1
It is observed that
since \(\sum_{j=0}^{\infty}p_{n,k}(x,c)=1\).
Let \(\operatorname {C}_{\mathrm{B}}[0,\infty)\) denote the space of all continuous and bounded functions on \([0,\infty)\), where the norm is defined by
Lemma 3
For every \(h\in \operatorname {C}_{\mathrm{B}}[0,\infty)\), we have
Lemma 3 can easily be proved using (2.1).
Remark 2
We observe that
in view of the inequality
Hence, from (1.4), we get
3 Main results
For \(x\in(0,\infty)\), \(t\in[0,\infty)\), and \(0< r\leq1\), as we can see in Özarslan and Duman [28], the Lipschitz type space is defined as
In the following theorem, we obtain the rate of convergence of the operators \(P_{n,\alpha}^{\beta,c}\) for functions in \(\operatorname {Lip}_{M}^{*}(r)\).
Theorem 1
Let \(h\in \operatorname {Lip}_{M}^{*}(r)\) and \(r\in(0,1]\). Then, for all \(x\in(0,\infty)\), we have
Proof
Using Remark 2, we get
Taking \(p=\frac{2}{r}\) and \(q=\frac{2}{2-r}\) and applying Hölder’s inequality, we obtain
Combining (3.1) and (3.2), we get
This completes the proof. □
In the following, we present some weighted approximation results. First, we recall some basic notations. Let \(\operatorname {B}_{2}[0,\infty)=\{h:[0, \infty)\to {\mathbb {R}}:\vert h(x)\vert\leq M_{h}(1+x^{2})\mbox{ for all }x\in [0,\infty) \}\). Further, let \(\operatorname {C}_{2}[0,\infty)\) be the subspace of \(\operatorname {B}_{2}[0,\infty)\) consisting of continuous functions defined on \([0,\infty)\). The norm in \(\operatorname {C}_{2}[0,\infty)\) is given by
Also, let
The next theorem provides us the degree of approximation of \(P_{n,\alpha}^{\beta,c}\) in terms of the classical modulus of continuity for the functions in the weighted space \(\operatorname {C}_{2}[0, \infty)\).
Theorem 2
For \(h\in \operatorname {C}_{2}[0,\infty)\), we have
where \(\omega_{b+1}(h;\delta)\) is the modulus of continuity of h on \([0,b+1]\).
Proof
From [29], for \(x\in[0,b]\) and \(t\geq0\), we obtain
Applying Remark 2 and the Cauchy–Schwarz inequality, we get
Choosing \(\delta=\sqrt{\mu_{n,2}^{\beta,c}(x)}\), we get the desired result. □
To determine the rate of convergence for functions in \(\operatorname {C}_{2}^{0}[0, \infty)\), Yüksel and Ispir [6] introduced the weighted modulus of continuity as
In the following lemma, we state the properties of the weighted modulus of continuity \(\Omega(h;\delta)\).
Lemma 4
(see [6])
Let \(h\in \operatorname {C}_{2}^{0}[0,\infty)\). Then the following results hold.
-
(1)
\(\Omega(h;\delta)\) is monotonically increasing in δ.
-
(2)
\(\lim_{\delta\to0^{+}}\Omega(h;\delta)=0\).
-
(3)
For each \(m\in {\mathbb {N}}\), \(\Omega(h;m\delta)\leq m\Omega(h;\delta)\).
-
(4)
For each \(\lambda\in[0,\infty)\), \(\Omega(h;\lambda\delta)\leq(1+ \lambda)\Omega(h;\delta)\).
Theorem 3
Let \(h\in \operatorname {C}_{2}^{0}[0,\infty)\), \(\beta=\beta(n)\to0\) as \(n\to\infty\) with \(\lim_{n\to\infty}n\beta=l\in {\mathbb {R}}\), and \(b>0\). Then
Proof
Let \(x_{0}\in[0,\infty)\) be arbitrary but fixed. Then
Since \(\vert h(x)\vert\leq \Vert h\Vert _{2}(1+x^{2})\), we have
Let \(\varepsilon >0\) be arbitrary. We choose \(x_{0}\) to be so large that
For \(\varepsilon >0\), there exists \(n_{1}\in {\mathbb {N}}\) such that
Hence, using (3.4), we get
Applying Theorem 2, we can find \(n_{2}\in {\mathbb {N}}\) such that
for all n greater than equal to \(n_{2}\). Combining (3.3)–(3.6), we obtain
This proves the required result. □
In the following theorem, we establish the rate of convergence of the operators \(P_{n,\alpha}^{\beta,c}\) in terms of the weighted modulus of continuity Ω.
Theorem 4
Let \(h\in \operatorname {C}_{2}^{0}[0,\infty)\). If \(\beta=\beta(n)\to0\) as \(n\to\infty\) and \(\lim_{n\to\infty}n\beta(n)=l\in {\mathbb {R}}\), then, for sufficiently large n, we have
where C is a positive constant independent of h and n.
Proof
For \(x\in(0,\infty)\) and \(\delta>0\), using the definition of weighted modulus of continuity and Lemma 4, we have
Applying \(P_{n,\alpha}^{\beta,c}(\cdot;x)\) to both sides of the above inequality, we can write
From Lemma 2, for sufficiently large n, it follows that
where C is a positive constant. Now, applying the Cauchy–Schwarz inequality in the last term of (3.7), we obtain
Combining estimates (3.7)–(3.9) and taking
we reach the required result. □
Now our aim is to discuss the rate of convergence in terms of the unified Ditzian–Totik modulus of smoothness \(\omega_{\phi^{\tau}}(h,t)\), \(0\leq\tau\leq1\). First, we define the Ditzian–Totik modulus of smoothness and the Peetre K-functional. Let \(\phi(x)=\sqrt{x(2+cx)}\) and \(h\in \operatorname {C}_{\mathrm{B}}[0,\infty)\). The modulus \(\omega_{\phi^{\tau}}(h,t)\), \(0\leq\tau\leq1\), is defined as
and the appropriate K-functional is given by
where \(W_{\tau}\) is the subspace of the space of locally absolutely continuous functions g on \([0,\infty)\), with \(\Vert \phi^{\tau }g'\Vert < \infty\). By [30, Theorem 2.1.1], there exists a constant \(N>0\) such that
Theorem 5
If \(h\in \operatorname {C}_{\mathrm{B}}[0,\infty)\), then
for sufficiently large n, where C is independent of h and n.
Proof
By the definition of \(K_{\phi^{\tau}}(h,t)\), there exists a function \(g\in W_{\tau}\) such that
We can write
Since \(g\in W_{\tau}\), we obtain
and so
By applying Hölder’s inequality, we get
Now
the inequality \(\vert a+b\vert^{\tau}\leq\vert a\vert^{\tau }+\vert b\vert^{\tau}\), \(0\leq\tau\leq1\), and (3.14) imply
Thus, from (3.13), (3.15), and the Cauchy–Schwarz inequality, we get
Note that for each \(x\in(0,\infty)\),
and thus, for \(\varepsilon >0\), there exists \(n_{0}\in {\mathbb {N}}\) such that
Choosing \(\varepsilon =(2+cx)^{-\tau}\), we obtain
Therefore, using (3.8) and (3.16), we get
for sufficiently large n. Thus, from (3.12), (3.17), and (3.11) (in that order), we find
where \(C'=\max\{ 2,2^{\tau}(1+\sqrt{2})\sqrt{\alpha C} \} \) and \(C=2C'\). By using relation (3.10), we reach the required result. □
Lastly, we obtain the convergence rate for functions having derivatives equivalent with a function of bounded variation. Let \(\operatorname {DBV}[0,\infty)\) be the class of functions \(h\in \operatorname {B}_{2}[0,\infty)\) having a derivative of bounded variation on every finite subinterval of \([0,\infty)\). The function \(h\in \operatorname {DBV}[0,\infty)\) has the representation
where j is a function of bounded variation on each finite subinterval of \([0,\infty)\). For this purpose, we use the following auxiliary result.
Lemma 5
For fixed \(u\in(0,\infty)\) and sufficiently large n, we have
and
where C is a positive constant.
Proof
Applying Remark 2 and using (3.8), we have
showing (3.19). Similarly, applying Remark 2 and using (3.8), we get
showing (3.20). □
Theorem 6
Let \(h\in \operatorname {DBV}[0,\infty)\). Then, for every \(x\in(0,\infty)\) and sufficiently large n, we have
where \(\bigvee_{a}^{b}h(x) \) represents the total variation of h on \([a,b]\), M is a constant, and \(h'_{x}\) is defined by
Proof
For any \(h\in \operatorname {DBV}[0,\infty)\), from (3.21), we may write
where
Since \(P_{n,\alpha}^{\beta,c}(1;x)=1\), using (1.4), for every \(x\in(0,\infty)\), we get
From (3.22) and (3.23), we get
where
and
Obviously,
Next, using (1.4), we get
and
Combining (3.23)–(3.26), applying Remark 2 and the Cauchy–Schwarz inequality, and using (3.8), we obtain
Now we estimate \(C_{n,\alpha}^{\beta,c}(h'_{x},x)\) and \(D_{n,\alpha }^{\beta,c}(h'_{x},x)\). Since
substituting \(y=x-x/\sqrt{n}\) and applying Lemma 5, we get
Substituting \(u=x/(x-t)\), we obtain
Thus,
Again, using the Cauchy–Schwarz inequality, integration by parts, and applying Lemma 5 to estimate \(D_{n,\alpha}^{\beta,c}(h'_{x},x)\), we get
\(D_{n,\alpha}^{\beta,c}(h'_{x},x)\) is estimated in a manner similar to \(C_{n,\alpha}^{\beta,c}(h'_{x},x)\). Putting \(t=x+x/u\) and using (3.8), we get
For \(t\geq2x\), we have \(t\leq2(t-x)\) and \(x\leq t-x\). Now, using (3.8), we obtain
Combining this with (3.27)–(3.29) yields the desired result. □
4 Conclusion
The Bézier variant of a sequence of mixed hybrid operators has been introduced and the rate of convergence by means of the Lipschitz class and the modulus of continuity has been established. The weighted approximation properties and a direct approximation theorem have been obtained. The approximation of functions with derivatives of bounded variation has been studied.
References
Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Comm. Soc. Math. Kharkov 13(2), 1–2 (1912)
Szász, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bur. Stand. 45(3), 239–245 (1950)
Jain, G.C.: Approximation of functions by a new class of linear operators. J. Aust. Math. Soc. 13, 271–276 (1972)
Gupta, V., Greubel, G.C.: Moment estimations of new Szász–Mirakyan–Durrmeyer operators. Appl. Math. Comput. 271, 540–547 (2015)
Gupta, V., Agarwal, R.P.: Convergence Estimates in Approximation Theory. Springer, Cham (2014)
Yüksel, I., Ispir, N.: Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 52(10–11), 1463–1470 (2007) 2006
Acu, A.M., Gupta, V.: Direct results for certain summation-integral type Baskakov–Szász operators. Results Math. 72(3), 1161–1180 (2017)
Phillips, R.S.: An inversion formula for Laplace transforms and semi-groups of linear operators. Ann. Math. (2) 59, 325–356 (1954)
Agrawal, P.N., Mohammad, A.J.: Linear combination of a new sequence of linear positive operators. Rev. Unión Mat. Argent. 44(1), 33–41 (2003)
Zeng, X.-M., Piriou, A.: On the rate of convergence of two Bernstein–Bézier type operators for bounded variation functions. J. Approx. Theory 95(3), 369–387 (1998)
Zeng, X.-M., Chen, W.: On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation. J. Approx. Theory 102(1), 1–12 (2000)
Zeng, X.-M., Tao, W.: Rate of convergence of the integral type Lupas–Bézier operators. Kyungpook Math. J. 43(4), 593–604 (2003)
Agrawal, P.N., Ispir, N., Kajla, A.: Approximation properties of Bezier-summation-integral type operators based on Polya–Bernstein functions. Appl. Math. Comput. 259, 533–539 (2015)
Agrawal, P.N., Mohammad, A.J.: On simultaneous approximation by a linear combination of a new sequence of linear positive operators. Turk. J. Math. 28(4), 353–365 (2004)
Aral, A.: A generalization of Szász–Mirakyan operators based on q-integers. Math. Comput. Model. 47(9–10), 1052–1062 (2008)
Chang, G.-Z.: Generalized Bernstein–Bezier polynomials. J. Comput. Math. 1(4), 322–327 (1983)
Goyal, M., Agrawal, P.N.: Bèzier variant of the generalized Baskakov Kantorovich operators. Boll. Unione Mat. Ital. 8(4), 229–238 (2016)
Guo, S.S., Liu, G.F., Song, Z.J.: Approximation by Bernstein–Durrmeyer–Bézier operators in \(L_{p}\) spaces. Acta Math. Sci. Ser. A Chin. Ed. 30(6), 1424–1434 (2010)
Gupta, V., Mohapatra, R.N.: Approximation by Durrmeyer–Bezier operators. Nonlinear Anal., Real World Appl. 9(4), 1491–1498 (2008)
Wang, P., Zhou, Y.: A new estimate on the rate of convergence of Durrmeyer–Bézier operators. J. Inequal. Appl. 2009, Article ID 702680 (2009)
Zeng, X.-M.: On the rate of convergence of two Bernstein–Bézier type operators for bounded variation functions. II. J. Approx. Theory 104(2), 330–344 (2000)
Acar, T., Agrawal, P.N., Neer, T.: Bezier variant of the Bernstein–Durrmeyer type operators. Results Math. 72(3), 1341–1358 (2017)
Bojanić, R., Chêng, F.H.: Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation. J. Math. Anal. Appl. 141(1), 136–151 (1989)
Gupta, V., Abel, U., Ivan, M.: Rate of convergence of beta operators of second kind for functions with derivatives of bounded variation. Int. J. Math. Math. Sci. 23, 3827–3833 (2005)
Gupta, V., Vasishtha, V., Gupta, M.K.: Rate of convergence of summation-integral type operators with derivatives of bounded variation. J. Inequal. Pure Appl. Math. 2003, Article ID 34 (2003)
Kajla, A., Acu, A.M., Agrawal, P.N.: Baskakov–Szász-type operators based on inverse Pólya–Eggenberger distribution. Ann. Funct. Anal. 8(1), 106–123 (2017)
Kajla, A., Goyal, M.: Blending type approximation by Bernstein–Durrmeyer type operators. Mat. Vesn. 70(1), 40–54 (2018)
Özarslan, M.A., Duman, O.: Local approximation behavior of modified SMK operators. Miskolc Math. Notes 11(1), 87–99 (2010)
İbikli, E., Gadjieva, E.A.: The order of approximation of some unbounded functions by the sequences of positive linear operators. Turk. J. Math. 19(3), 331–337 (1995)
Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer Series in Computational Mathematics, vol. 9. Springer, New York (1987)
Acknowledgements
The last author expresses her sincere thanks to Ms. Sheetal Deshwal and Ms. Tarul Garg for their continued support and helpful discussions during the preparation of the paper.
Author information
Authors and Affiliations
Contributions
All authors have equally contributed to this work. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
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
Agrawal, P.N., Araci, S., Bohner, M. et al. Approximation degree of Durrmeyer–Bézier type operators. J Inequal Appl 2018, 29 (2018). https://doi.org/10.1186/s13660-018-1622-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1622-1