Abstract and Applied Analysis

Volume 2014 (2014), Article ID 959586, 11 pages

http://dx.doi.org/10.1155/2014/959586

Research Article

Approximation by Genuine -Bernstein-Durrmeyer Polynomials in Compact Disks in the Case

Nazim I. Mahmudov

Department of Mathematics, Eastern Mediterranean University, Gazimagusa, TRNC, Via Mersin 10, Turkey

Received 10 January 2014; Accepted 2 February 2014; Published 16 March 2014

Academic Editor: Sofiya Ostrovska

Copyright © 2014 Nazim I. Mahmudov. 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 -generalization of the genuine Bernstein-Durrmeyer polynomials in the case , which are no longer positive linear operators on . Quantitative estimates of the convergence, the Voronovskaja-type theorem, and saturation of convergence for complex genuine -Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that, for functions analytic in , , the rate of approximation by the genuine -Bernstein-Durrmeyer polynomials is of order versus for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine -Bernstein-Durrmeyer for . This paper represents an answer to the open problem initiated by Gal in (2013, page 115).

1. Introduction

In several recent papers, convergence properties of complex -Bernstein polynomials, proposed by Phillips [1], attached to an analytic function in closed disks, were intensively studied. Ostrovska [2, 3] and Wang and Wu [4, 5] have investigated convergence properies of in the case . In the case , the -Bernstein polynomials are no longer positive operators; however, for a function analytic in a disc , it was proved in [2] that the rate of convergence of to has the order (versus for the classical Bernstein polynomials). Moreover, Ostrovska [3] obtained Voronovskaya-type theorem for monomials. If , then qualitative Voronovskaja-type theorem and saturation results for complex -Bernstein polynomials were obtained by Wang and Wu [4]. Wu [5] studied saturation of convergence on the interval for the -Bernstein polynomials of a continuous function for arbitrary fixed .

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 Păltănea [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 -Bernstein-Durrmeyer operators () 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 -Durrmeyer-type operators () in the real case was first initiated by Derriennic [19].

Also, for the case , exact quantitative estimates and quantitative Voronovskaja-type results for complex -Lorentz polynomials, -Stancu polynomials [20], -Stancu-Faber polynomials, -Bernstein-Faber polynomials, -Kantorovich polynomials [21], -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 -Durrmeyer-kind operators for presented an open problem. This paper presents a positive solution to this problem.

In this paper we define the genuine -Bernstein-Durrmeyer polynomials for . Note that similar to the -Bernstein operators the genuine -Bernstein-Durrmeyer operators in the case are not positive operators on . The lack of positivity makes the investigation of convergence in the case essentially more difficult than that for . 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 -Bernstein-Durrmeyer polynomials. Our results show that approximation properties of the complex genuine -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 -calculus; see, for example, [24, 25]. Let . For any , the -integer is defined by

and the -factorial is defined by

For integers , the -binomial is defined by

For we obviously get , , and . Moreover

For fixed , , we denote the -derivative of by

The -analogue of integration in the interval (see [24]) is defined by

Let be a disc in the complex plane . Denote by the space of all analytic functions on . For we assume that for all . The norm . We denote for all .

Definition 1. For , the genuine -Bernstein-Durrmeyer operator is defined as follows:

where for the sum is empty; that is, it is equal to .

are linear operators reproducing linear functions and interpolating every function at and . The genuine -Bernstein-Durrmeyer operators are positive operators on for , and they are not positive for . As a consequence, the cases and are not similar to each other regarding the convergence. For and we recapture the classical () genuine Bernstein-Durrmeyer polynomials.

We start with the following quantitative estimates of the convergence for complex -Bernstein-Durrmeyer polynomials attached to an analytic function in a disk of radius and center .

Theorem 2. Let , , and . Then for all one has

Theorem 2 says that, for functions analytic in , , the rate of approximation by the genuine -Bernstein-Durrmeyer polynomials () is of order versus 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:

and, for all , . For the complex genuine -Bernstein-Durrmeyer () a quantitative estimate is obtained by Gal [13] () and Mahmudov [16] () in the following form:

and, for all , .

To formulate and prove the Voronovskaja-type theorem with a quantitative estimate in the case we introduce a function .

Let and let . For , we define

And, for ,

The next theorem gives Voronovskaja-type result in compact disks; for complex -Bernstein-Durrmeyer polynomials attached to an analytic function in , and center in terms of the function .

Theorem 3. Let , , and . The following Voronovskaja-type result holds:

For all , .

Now we are in position to prove that the order of approximation in Theorem 2 is exactly versus for the classical genuine Bernstein-Durrmeyer polynomials; see [13].

Theorem 4. Let , , and . If is not a polynomial of degree ≤1, the estimate,

holds, where the constant depends on , , and but is independent of .

From Theorem 3 we conclude that, for , in and therefore . Furthermore, we have the following saturation of convergence for the genuine -Bernstein-Durrmeyer polynomials for fixed .

Theorem 5. Let , . If a function is analytic in the disc , then for infinite number of points having an accumulation point on if and only if is linear.

The next theorem shows that , is continuous in the parameter for , .

Theorem 6. Let and . Then, for any ,

uniformly on .

3. Auxiliary Results

The -analogue of beta function for (see [24]) is defined as

Since we consider the case , we need to use as follows:

Also, it is known that

For , we have

Thus, we get the following formula for :

Note that, for , we have

Lemma 7. is a polynomial of degree less than or equal to and

Proof. From (20) it follows that

Now using

where , , are the constants independent of , we get

Since is a polynomial of degree less than or equal to and , , it follows that is a polynomial of degree less than or equal to .

Lemma 8. The numbers , given by (24), enjoy the following properties:

Also, the following lemma holds.

Lemma 9. For all the identity,

holds.

Proof. It follows from end points interpolation property of and . Indeed

Lemma 9 implies that for all and we have

For our purpose first we need a recurrence formula for .

Lemma 10. For all and one has

Proof. By simple calculation we obtain (see [27])

It follows that

which implies the recurrence in the statement.

Let

Using the recurrence formula (30) we prove two more recurrence formulas.

Lemma 11. For all and one has

where

Proof. From the recurrence formula in Lemma 10, for all , we get

where

Again by simple calculation we obtain

where and can be simplified as follows:

Lemma 12. Let and . The function has the following representation:

Proof. Using the following identity:

we get

where .

4. Proofs of the Main Results

Firstly we prove that . Indeed denoting with , by the linearity of , we have

and it is sufficient to show that, for any fixed and with , we have . But this is immediate from , the norm being defined as , and from the inequality

valid for all , where

Therefore we get

as and .

Proof of Theorem 2. From the recurrence formula (34) and the inequality (29) for we get

It is known that, by a linear transformation, the Bernstein inequality in the closed unit disk becomes

which, combined with the mean value theorem in complex analysis, implies

for all , where is a complex polynomial of degree ≤. It follows that

By writing the last inequality for , we easily obtain, step by step, the following:

It follows that

The second main result of the paper is the Voronovskaja-type theorem with a quantitative estimate for the complex version of genuine -Bernstein-Durrmeyer polynomials.

Proof of Theorem 3. By Lemma 11 we have

where

It follows that

for all , , and . Equation (54) implies that for

By writing the last inequality for , we easily obtain, step by step, the following:

Proof of Theorem 4. For all and we get

It follows that

Because by hypothesis is not a polynomial of degree in , it follows . Indeed, assuming the contrary it follows that for all ; that is, for all . Thus and is linear, which is a contradiction with the hypothesis.

Now, by Theorem 3, we have

Consequently, there exists (depending only on and ) such that for all we have

which implies that

For we have

which finally implies that

for all , with , which ends the proof.

Proof of Theorem 6. Let , be fixed. Then, by Lemma 12 for any and , we have

Using the inequality

we get, for and ,

Since , we can find that such that

Thus, for sufficiently close to from the right, we conclude that

uniformly on . The proof is finished.

Proof of Theorem 5. Then, by Theorem 3, we get for infinite number of points having an accumulation point on . Since , by the unicity Theorem for analytic functions, we get in , and, therefore, by (11), , . Thus, is linear. Theorem 5 is proved.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author dedicates this paper to Professor Agamirza E. Bashirov at his 60th anniversary.

References

G. M. Phillips, “A survey of results on the -Bernstein polynomials,” IMA Journal of Numerical Analysis, vol. 30, no. 1, pp. 277–288, 2010.

S. Ostrovska, “-Bernstein polynomials and their iterates,” Journal of Approximation Theory, vol. 123, no. 2, pp. 232–255, 2003.

S. Ostrovska, “The sharpness of convergence results for -Bernstein polynomials in the case ,” Czechoslovak Mathematical Journal, vol. 58, no. 4, pp. 1195–1206, 2008.

H. Wang and X. Wu, “Saturation of convergence for -Bernstein polynomials in the case ,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 744–750, 2008.

Z. Wu, “The saturation of convergence on the interval for the -Bernstein polynomials in the case ,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 137–141, 2009.

W. Chen, “On the modified Bernstein-Durrmeyer operator,” in Proceedings of the 5th Chinese Conference on Approximation Theory, Zhengzhou, China, 1987.

T. N. T. Goodman and A. Sharma, “A Bernstein type operator on the simplex,” Mathematica Balkanica, vol. 5, no. 2, pp. 129–145, 1991.

H. Gonska, D. Kacsó, and I. Raşa, “On genuine Bernstein-Durrmeyer operators,” Results in Mathematics, vol. 50, no. 3-4, pp. 213–225, 2007.

P. E. Parvanov and B. D. Popov, “The limit case of Bernstein's operators with Jacobi-weights,” Mathematica Balkanica, vol. 8, no. 2-3, pp. 165–177, 1994.

T. Sauer, “The genuine Bernstein-Durrmeyer operator on a simplex,” Results in Mathematics, vol. 26, no. 1-2, pp. 99–130, 1994.

S. Waldron, “A generalised beta integral and the limit of the Bernstein-Durrmeyer operator with Jacobi weights,” Journal of Approximation Theory, vol. 122, no. 1, pp. 141–150, 2003.

R. Păltănea, Approximation Theory Using Positive Linear Operators, Birkhäuser, Boston, Mass, USA, 2004.

S. G. Gal, “Approximation by complex genuine Durrmeyer type polynomials in compact disks,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 1913–1920, 2010.

G. A. Anastassiou and S. G. Gal, “Approximation by complex Bernstein-Durrmeyer polynomial in compact disks,” Mediterranean Journal of Mathematics, vol. 7, no. 4, pp. 471–482, 2010.

S. G. Gal, “Approximation by complex Bernstein-Durrmeyer polynomials with Jacobi weights in compact disks,” Mathematica Balkanica, vol. 24, no. 1-2, pp. 103–119, 2010.

N. I. Mahmudov, “Approximation by genuine -Bernstein-Durrmeyer polynomials in compact disks,” Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 1, pp. 77–89, 2011.

N. I. Mahmudov, “Approximation by Bernstein-Durrmeyer-type operators in compact disks,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1231–1238, 2011.

S. G. Gal, V. Gupta, and N. I. Mahmudov, “Approximation by a complex q-Durrmeyer type operator,” Annali dell'Universitá di Ferrara, vol. 58, no. 1, pp. 65–87, 2012.

M.-M. Derriennic, “Modified Bernstein polynomials and Jacobi polynomials in -calculus,” Rendiconti del Circolo Matematico di Palermo II, vol. 76, supplement, pp. 269–290, 2005.

N. I. Mahmudov, “Convergence properties and iterations for -Stancu polynomials in compact disks,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3763–3769, 2010.

N. I. Mahmudov and M. Kara, “Approximation theorems for generalized complex Kantorovich-type operators,” Journal of Applied Mathematics, vol. 2012, Article ID 454579, 14 pages, 2012.

N. I. Mahmudov, “Approximation properties of complex -Szász-Mirakjan operators in compact disks,” Computers & Mathematics with Applications, vol. 60, no. 6, pp. 1784–1791, 2010.

S. G. Gal, Overconvergence in Complex Approximation, Springer, New York, NY, USA, 2013.

G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1999.

A. Aral, V. Gupta, and R. P. Agarwal, Applications of q-Calculus in Operator Theory, Springer, New York, NY, USA, 2013.

H. Gonska, P. Piţul, and I. Raşa, “On Peano's form of the Taylor remainder, Voronovskaja's theorem and the commutator of positive linear operators,” pp. 55–80, NAAT, Cluj-Napoca, Romania, 2006.

N. I. Mahmudov, “The moments for -Bernstein operators in the case ,” Numerical Algorithms, vol. 53, no. 4, pp. 439–450, 2010.