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Academic Editor:Sofiya Ostrovska
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, TRNC, Via Mersin 10, Turkey
Received 10 January 2014; Accepted 2 February 2014; 16 March 2014
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In several recent papers, convergence properties of complex q -Bernstein polynomials, proposed by Phillips [1], attached to an analytic function f in closed disks, were intensively studied. Ostrovska [2, 3] and Wang and Wu [4, 5] have investigated convergence properies of Bn,q in the case q>1 . In the case q>1 , the q -Bernstein polynomials are no longer positive operators; however, for a function analytic in a disc ...R :={z∈...:|z|<R}, R>q , it was proved in [2] that the rate of convergence of {Bn,q (f;z)} to f(z) has the order q-n (versus 1/n for the classical Bernstein polynomials). Moreover, Ostrovska [3] obtained Voronovskaya-type theorem for monomials. If q...5;1 , then qualitative Voronovskaja-type theorem and saturation results for complex q -Bernstein polynomials were obtained by Wang and Wu [4]. Wu [5] studied saturation of convergence on the interval [0,1] for the q -Bernstein polynomials of a continuous function f for arbitrary fixed q>1 .
Genuine Bernstein-Durrmeyer operators were first considered by Chen [6] and Goodman and Sharma [7] around 1987. In recent years, the genuine Bernstein-Durrmeyer operators have been investigated intensively by a number of authors. Among the many papers written on the genuine Bernstein-Durrmeyer operators, we mention here only the ones by Gonska et al. [8], Parvanov and Popov [9], Sauer [10], Waldron [11], and the book of Paltanea [12].
On the other hand, Gal [13] obtained quantitative estimates of the convergence and of the Voronovskaja-type theorem in compact disks, for the complex genuine Bernstein-Durrmeyer polynomials attached to analytic functions. Besides, in other very recent papers, similar studies were done for complex Bernstein-Durrmeyer operators in Anastassiou and Gal [14], for complex Bernstein-Durrmeyer operators based on Jacobi weights in Gal [15], for complex genuine q -Bernstein-Durrmeyer operators (0<q<1 ) by Mahmudov [16], and for other kinds of complex Durrmeyer operators in Mahmudov [17] and Gal et al. [18]. It should be stressed out that study of q -Durrmeyer-type operators (0<q<1 ) in the real case was first initiated by Derriennic [19].
Also, for the case q>1 , exact quantitative estimates and quantitative Voronovskaja-type results for complex q -Lorentz polynomials, q -Stancu polynomials [20], q -Stancu-Faber polynomials, q -Bernstein-Faber polynomials, q -Kantorovich polynomials [21], q -Szász-Mirakjan operators [22] obtained by different researchers are collected in the recent book of Gal [23]. In this book the definition and study of complex q -Durrmeyer-kind operators for q>1 presented an open problem. This paper presents a positive solution to this problem.
In this paper we define the genuine q -Bernstein-Durrmeyer polynomials for q>1 . Note that similar to the q -Bernstein operators the genuine q -Bernstein-Durrmeyer operators in the case q>1 are not positive operators on C[0,1] . The lack of positivity makes the investigation of convergence in the case q>1 essentially more difficult than that for 0<q<1 . We present upper estimates in approximation and we prove the Voronovskaja-type convergence theorem in compact disks in ... , centered at origin, with quantitative estimate of this convergence. These results allow us to obtain the exact degrees of approximation by complex genuine q -Bernstein-Durrmeyer polynomials. Our results show that approximation properties of the complex genuine q -Bernstein-Durrmeyer polynomials are better than approximation properties of the complex Bernstein-Durrmeyer polynomials considered in [13].
2. Main Results
We begin with some notations and definitions of q -calculus; see, for example, [24, 25]. Let q>0 . For any n∈...∪{0} , the q -integer [n]q is defined by [figure omitted; refer to PDF] and the q -factorial [n]q ! is defined by [figure omitted; refer to PDF] For integers 0...4;k...4;n , the q -binomial is defined by [figure omitted; refer to PDF] For q=1 we obviously get [n]q =n , [n]q !=n! , and [nk]q =(nk) . Moreover [figure omitted; refer to PDF]
For fixed q>0 , q...0;1 , we denote the q -derivative Dq f(z) of f by [figure omitted; refer to PDF]
The q -analogue of integration in the interval [0,A] (see [24]) is defined by [figure omitted; refer to PDF] Let ...R be a disc ...R :={z∈...:|z|<R} in the complex plane ... . Denote by H(...R ) the space of all analytic functions on ...R . For f∈H(...R ) we assume that f(z)=∑m=0∞ ...amzm for all z∈...R . The norm ||f||r :=max...{|f(z)|:|z|...4;r} . We denote em (z)=zm for all m∈...∪{0} .
Definition 1.
For f:[0,1][arrow right]... , the genuine q -Bernstein-Durrmeyer operator is defined as follows: [figure omitted; refer to PDF] where for n=1 the sum is empty; that is, it is equal to 0 .
U n , q ( f ; z ) are linear operators reproducing linear functions and interpolating every function f∈C[0,1] at 0 and 1 . The genuine q -Bernstein-Durrmeyer operators are positive operators on C[0,1] for 0<q...4;1 , and they are not positive for q>1 . As a consequence, the cases 0<q...4;1 and q>1 are not similar to each other regarding the convergence. For q[arrow right]1- and q[arrow right]1+ we recapture the classical (q=1 ) genuine Bernstein-Durrmeyer polynomials.
We start with the following quantitative estimates of the convergence for complex q -Bernstein-Durrmeyer polynomials attached to an analytic function in a disk of radius R>1 and center 0 .
Theorem 2.
Let f∈H(...R ) , 1...4;r<R/q , and q>1 . Then for all |z|...4;r one has [figure omitted; refer to PDF]
Theorem 2 says that, for functions analytic in ...R , R>q , the rate of approximation by the genuine q -Bernstein-Durrmeyer polynomials (q>1 ) is of order q-n versus 1/n for the classical genuine Bernstein-Durrmeyer polynomials; see [13].
The Voronovskaja theorem for the real case with a quantitative estimate is obtained by Gonska et al. [26] in the following form: [figure omitted; refer to PDF] and, for all n∈... , 0...4;x...4;1 . For the complex genuine q -Bernstein-Durrmeyer (0<q...4;1 ) a quantitative estimate is obtained by Gal [13] (q=1 ) and Mahmudov [16] (0<q<1 ) in the following form: [figure omitted; refer to PDF] and, for all n∈... , |z|...4;r .
To formulate and prove the Voronovskaja-type theorem with a quantitative estimate in the case q>1 we introduce a function Lq (f;z) .
Let R>q...5;1 and let f∈H(...R ) . For |z|<R/q2 , we define [figure omitted; refer to PDF] And, for 0<q...4;1 , [figure omitted; refer to PDF]
The next theorem gives Voronovskaja-type result in compact disks; for complex q -Bernstein-Durrmeyer polynomials attached to an analytic function in ...R , R>q2 >1 and center 0 in terms of the function Lq (f;z) .
Theorem 3.
Let f∈H(...R ) , 1...4;r<R/q2 , and q>1 . The following Voronovskaja-type result holds: [figure omitted; refer to PDF] For all n∈... , |z|...4;r .
Now we are in position to prove that the order of approximation in Theorem 2 is exactly q-n versus 1/n for the classical genuine Bernstein-Durrmeyer polynomials; see [13].
Theorem 4.
Let 1<q<R , 1...4;r<R/q2 , and f∈H(...R ) . If f is not a polynomial of degree ...4;1, the estimate, [figure omitted; refer to PDF] holds, where the constant Cr,q (f) depends on f , q , and r but is independent of n .
From Theorem 3 we conclude that, for q>1 , [n+1]q (Un,q (f;z)-f(z))[arrow right]Lq (f;z) in H(...R/q2 ) and therefore Lq (f;z)∈H(...R/q2 ) . Furthermore, we have the following saturation of convergence for the genuine q -Bernstein-Durrmeyer polynomials for fixed q>1 .
Theorem 5.
Let 1<q<R , 1...4;r<R/q2 . If a function f is analytic in the disc ...R/q2 , then |Un,q (f;z)-f(z)|=o(q-n ) for infinite number of points having an accumulation point on ...R/q2 if and only if f is linear.
The next theorem shows that Lq (f;z), q...5;1 , is continuous in the parameter q for f∈H(...R ) , R>1 .
Theorem 6.
Let R>1 and f∈H(...R ) . Then, for any r, 0<r<R , [figure omitted; refer to PDF] uniformly on ...R .
3. Auxiliary Results
The q -analogue of beta function for 0<q<1 (see [24]) is defined as [figure omitted; refer to PDF] Since we consider the case q>1 , we need to use Bq-1 (m,n) as follows: [figure omitted; refer to PDF] Also, it is known that [figure omitted; refer to PDF] For m=0,1,... , we have [figure omitted; refer to PDF] Thus, we get the following formula for Un,q (em ;z) : [figure omitted; refer to PDF] Note that, for m=0,1,2 , we have [figure omitted; refer to PDF]
Lemma 7.
U n , q ( e m ; z ) is a polynomial of degree less than or equal to min...(m,n) and [figure omitted; refer to PDF]
Proof.
From (20) it follows that [figure omitted; refer to PDF] Now using [figure omitted; refer to PDF] where Sq (m,s)>0 , s=1,2,...,m , are the constants independent of k , we get [figure omitted; refer to PDF] Since Bn,q (es ;z) is a polynomial of degree less than or equal to min...(s,n) and Sq (m,s)>0 , s=1,2,...,m , it follows that Un,q (em ;z) is a polynomial of degree less than or equal to min...(m,n) .
Lemma 8.
The numbers Sq (m,s), (m,s)∈(...∪{0})×(...∪{0}) , given by (24), enjoy the following properties: [figure omitted; refer to PDF]
Also, the following lemma holds.
Lemma 9.
For all m,n∈... the identity, [figure omitted; refer to PDF] holds.
Proof.
It follows from end points interpolation property of Un,q (em ;z) and Bn,q (es ;z) . Indeed [figure omitted; refer to PDF]
Lemma 9 implies that for all m,n∈... and |z|...4;r we have [figure omitted; refer to PDF]
For our purpose first we need a recurrence formula for Un,q (em ;z) .
Lemma 10.
For all m,n∈...∪{0} and z∈... one has [figure omitted; refer to PDF]
Proof.
By simple calculation we obtain (see [27]) [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] which implies the recurrence in the statement.
Let [figure omitted; refer to PDF] Using the recurrence formula (30) we prove two more recurrence formulas.
Lemma 11.
For all m,n∈... and z∈... one has [figure omitted; refer to PDF] where [figure omitted; refer to PDF]
Proof.
From the recurrence formula in Lemma 10, for all m...5;2 , we get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Again by simple calculation we obtain [figure omitted; refer to PDF] where Tn,m1 (q) and Tn,m2 (q) can be simplified as follows: [figure omitted; refer to PDF]
Lemma 12.
Let q>1 and f∈H(...R ) . The function Lq (f;z) has the following representation: [figure omitted; refer to PDF]
Proof.
Using the following identity: [figure omitted; refer to PDF] we get [figure omitted; refer to PDF] where f(z)=∑m=0∞ ...amzm .
4. Proofs of the Main Results
Firstly we prove that Un,q (f;z)=∑m=0∞ ...amUn,q (em ,z) . Indeed denoting fk (z)=∑j=0k ...ajzj , |z|...4;r with m∈... , by the linearity of Un,q , we have [figure omitted; refer to PDF] and it is sufficient to show that, for any fixed n∈... and |z|...4;r with r...5;1 , we have lim...k[arrow right]∞Un,q (fk ,z)=Un,q (f;z) . But this is immediate from lim...k[arrow right]∞||fk -f||r =0 , the norm being defined as ||f||r =max...{|f(z)|:|z|...4;r} , and from the inequality [figure omitted; refer to PDF] valid for all |z|...4;r , where [figure omitted; refer to PDF] Therefore we get [figure omitted; refer to PDF] as Un,q (e0 ,z)=e0 (z) and Un,q (e1 ,z)=e1 (z) .
Proof of Theorem 2.
From the recurrence formula (34) and the inequality (29) for m...5;2 we get [figure omitted; refer to PDF] It is known that, by a linear transformation, the Bernstein inequality in the closed unit disk becomes [figure omitted; refer to PDF] which, combined with the mean value theorem in complex analysis, implies [figure omitted; refer to PDF] for all |z|...4;qr , where Pk (z) is a complex polynomial of degree ...4;k . It follows that [figure omitted; refer to PDF] By writing the last inequality for m=2,3,... , we easily obtain, step by step, the following: [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF]
The second main result of the paper is the Voronovskaja-type theorem with a quantitative estimate for the complex version of genuine q -Bernstein-Durrmeyer polynomials.
Proof of Theorem 3.
By Lemma 11 we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] for all m...5;2 , n∈... , and z∈... . Equation (54) implies that for |z|...4;r [figure omitted; refer to PDF] By writing the last inequality for m=3,4,... , we easily obtain, step by step, the following: [figure omitted; refer to PDF]
Proof of Theorem 4.
For all z∈...R and n∈... we get [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] Because by hypothesis f is not a polynomial of degree ...4;1 in ...R , it follows ||Lq (f;z)||r >0 . Indeed, assuming the contrary it follows that Lq (f;z)=0 for all z∈...r ¯ ; that is, Dq f(z)=Dq-1 f(z) for all z∈...r ¯ . Thus am =0, m=2,3,... and f is linear, which is a contradiction with the hypothesis.
Now, by Theorem 3, we have [figure omitted; refer to PDF] Consequently, there exists n1 (depending only on f and r ) such that for all n...5;n1 we have [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] For 1...4;n...4;n1 -1 we have [figure omitted; refer to PDF] which finally implies that [figure omitted; refer to PDF] for all n , with Cr,q (f)=min...{Mr,1,q (f),...,Mr,n1 -1,q (f),(1/2)||Lq (f;z)||r } , which ends the proof.
Proof of Theorem 6.
Let 1...4;r<R , 1<q0 <R/r be fixed. Then, by Lemma 12 for any 1...4;q...4;q0 and |z|...4;r , we have [figure omitted; refer to PDF] Using the inequality [figure omitted; refer to PDF] we get, for 1...4;q...4;q0 and |z|...4;r , [figure omitted; refer to PDF] Since f∈H(...R ) , we can find that N=N[varepsilon] ∈... such that [figure omitted; refer to PDF] Thus, for q sufficiently close to 1 from the right, we conclude that [figure omitted; refer to PDF] uniformly on ...r . The proof is finished.
Proof of Theorem 5.
Then, by Theorem 3, we get Lq (f;z)=lim...n[arrow right]∞[n+1]q (Un,q (f;z)-f(z))=0 for infinite number of points having an accumulation point on ...R/q2 . Since Lq (f;z)∈H(...R/q2 ) , by the unicity Theorem for analytic functions, we get Lq (f;z)=0 in ...R/q2 , and, therefore, by (11), am =0 , m=2,3,... . Thus, f is linear. Theorem 5 is proved.
Acknowledgment
The author dedicates this paper to Professor Agamirza E. Bashirov at his 60th anniversary.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2014 Nazim I. Mahmudov. Nazim I. Mahmudov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper deals with approximating properties of the newly defined q -generalization of the genuine Bernstein-Durrmeyer polynomials in the case q>1 , which are no longer positive linear operators on C0,1 . Quantitative estimates of the convergence, the Voronovskaja-type theorem, and saturation of convergence for complex genuine q -Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that, for functions analytic in z∈...:z<R , R>q , the rate of approximation by the genuine q -Bernstein-Durrmeyer polynomials q>1 is of order [superscript]q-n[/superscript] versus 1/n for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine q -Bernstein-Durrmeyer for q>1 . This paper represents an answer to the open problem initiated by Gal in (2013, page 115).
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer