AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 521709 10.1155/2014/521709 521709 Research Article The Rate of Convergence of Lupas q -Analogue of the Bernstein Operators http://orcid.org/0000-0002-8398-023X Wang Heping 1 Zhang Yanbo 2 Ostrovska Sofiya 1 School of Mathematical Sciences BCMIIS Capital Normal University Beijing 100048 China cnu.edu.cn 2 Department of Basic Courses Shandong Modern Vocational College Jinan Shandong 250104 China uxd.com.cn 2014 942014 2014 24 12 2013 12 01 2014 9 4 2014 2014 Copyright © 2014 Heping Wang and Yanbo Zhang. 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.

We discuss the rate of convergence of the Lupas q -analogues of the Bernstein operators R n , q ( f ; x ) which were given by Lupas in 1987. We obtain the estimates for the rate of convergence of R n , q ( f ) by the modulus of continuity of f , and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

1. Introduction

In 1912, Bernstein (see ) defined the Bernstein polynomials. Later, it was found that the Bernstein polynomials possess many remarkable properties, which made them an area of intensive research. Due to the development of q -calculus, generalizations of Bernstein polynomials connected with q -calculus have emerged. The first person to make progress in this direction was Lupas, who introduced a q -analogue of the Bernstein operator R n , q ( f ; x ) and investigated its approximating and shape-preserving properties in 1987 (see ). If q = 1 , then { R n , 1 ( f ; x ) } are the classical Bernstein polynomials. For q 1 , the operators R n , q ( f ; x ) are rational functions rather than polynomials. Other generalizations of the Bernstein polynomials, for example, the q -Bernstein polynomials (see ), the two-parametric generalization of q -Bernstein polynomials (see ), and the q -Bernstein-Durrmeyer operator (see ), had also been considered in recent years. Among these generalizations, q -Bernstein polynomials proposed by Phillips attracted the most attention and were studied widely by a number of authors (see [3, 615]). The Lupas q -analogues of the Bernstein operators { R n , q ( f ; x ) } are less known; see [2, 1621]. However, they have an advantage of generating positive linear operators for all q > 0 , whereas q -Bernstein polynomials generate positive linear operators only if q ( 0,1 ) .

In this paper, we will study the rate of convergence of the Lupas q -analogues of the Bernstein operators { R n , q ( f ; x ) } . We will obtain the estimates for the rate of convergence of R n , q ( f ) by the modulus of continuity of f , and show that the estimates are sharp in the sense of order for Lipschitz continuous functions. Our results demonstrate that the estimates for the rate of convergence of { R n , q ( f ; x ) } are essentially different from those for the classical Bernstein polynomials; however, they are very similar to those for the q -Bernstein polynomials in the case q ( 0,1 ) .

Throughout the paper, we always assume that f is a continuous real function on [ 0,1 ] , q > 0 , q 1 . Denote by C [ 0,1 ] (or C n [ 0,1 ] , 1 n ) the space of all continuous (correspondingly, n times continuously differentiable) real-valued functions on [ 0,1 ] equipped with the uniform norm · . The expression A ( n ) B ( n ) means that A ( n ) B ( n ) and A ( n ) B ( n ) , and A ( n ) B ( n ) means that there exists a positive constant c independent of n such that A ( n ) c B ( n ) .

To formulate our results, we need the following definitions.

Let q > 0 . For each nonnegative integer k , the q -integer [ k ] and the q -factorial [ k ] ! are defined by (1) [ k ] : = [ k ] q : = { ( 1 - q k ) ( 1 - q ) , q 1 k , q = 1 , [ k ] ! : = { [ k ] [ k - 1 ] [ 1 ] , k 1 1 , k = 0 . For integers 0 k n , the q -binomial coefficient is defined by (2) [ n k ] : = [ n ] ! [ k ] ! [ n - k ] ! .

In , Lupas proposed the q -analogue of the Bernstein operator R n , q ( f ; x ) : for each positive integer n , and f C [ 0,1 ] , (3) R n , q ( f , x ) : = { k = 0 n f ( [ k ] [ n ] ) r n , k ( q , x ) , 0 x < 1 f ( 1 ) , x = 1 , where (4) r n , k ( q ; x ) : = [ n k ] q k ( k - 1 ) / 2 x k ( 1 - x ) n - k ( 1 - x + q x ) ( 1 - x + q n - 1 x ) = [ n k ] q k ( k - 1 ) / 2 ( x / ( 1 - x ) ) k j = 0 n - 1 ( 1 + q j ( x / ( 1 - x ) ) ) .

In , Ostrovska proved that, for each f C [ 0,1 ] and q ( 0,1 ) , the sequence { R n , q ( f , x ) } converges to the limit operator R , q ( f , x ) uniformly on [ 0,1 ] as n , where (5) R , q ( f , x ) = { k = 0 f ( 1 - q k ) r k ( q ; x ) , 0 x < 1 f ( 1 ) , x = 1 , (6) r , k ( q ; x ) : = q k ( k - 1 ) / 2 ( x / ( 1 - x ) ) k ( 1 - q ) k [ k ] ! j = 0 ( 1 + q j ( x / ( 1 - x ) ) ) . When q > 1 , the following relations (see ) allow us to reduce to the case q ( 0,1 ) : (7) R n , q ( f ; x ) = R n , 1 / q ( g ; 1 - x ) , R , q ( f ; x ) = R , 1 / q ( g ; 1 - x ) , where g ( x ) = f ( 1 - x ) C [ 0,1 ] .

The problem to find the rate of convergence occurs naturally and this paper deals with the problem of finding estimates for the rate of convergence for a sequence of the q -analogue of the Bernstein operator R n , q ( f ; x ) for 0 < q < 1 . For f C [ 0,1 ] ,    t > 0 , the modulus of continuity ω ( f , t ) and the second modulus of smoothness ω 2 ( f , t ) are defined as follows: (8) ω ( f ; t ) : = sup | x - y | t x , y [ 0,1 ] | f ( x ) - f ( y ) | ; ω 2 ( f , t ) : = sup 0 < h t sup x [ 0,1 - 2 h ] | f ( x + 2 h ) - 2 f ( x + h ) + f ( x ) | .

The main results of the paper are as follows.

Theorem 1.

Let q ( 0,1 ) and let f C [ 0,1 ] . Then (9) R n , q ( f ) - R , q ( f ) C q ω ( f ; q n ) , where C q = 2 + 6 / ( 1 - q ) . This estimate is sharp in the following sense of order: for each α ,    0 < α 1 , there exists a function f α ( x ) which belongs to the Lipschitz class L i p α : = { f C [ 0,1 ] ω ( f ; t ) t α } such that (10) R n , q ( f α ) - R , q ( f α ) q n α .

Theorem 2.

Let 0 < q < 1 . Then (11) R n , q ( f ) - R , q ( f ) c ω 2 ( f ; q n ) . Furthermore, (12) sup 0 < q < 1 | R n , q ( f ) - R , q ( f ) | c ω 2 ( f ; n - 1 / 2 ) , where c is an absolute constant.

Remark 3.

From (12), it follows that, for each f C [ 0,1 ] , (13) lim n R n , q ( f ; x ) = R , q ( f ; x ) uniformly not only in x [ 0,1 ] , and but also in q ( 0,1 ] , which generalizes the Ostrovska's result in .

Remark 4.

It should be emphasized that Theorem 1 cannot be obtained in a way similar to the proof of the Popoviciu Theorem for the classical Bernstein polynomials (see ). It requires different estimation techniques due to the infinite product involved. Also, the proof in the paper is more difficult than the one used for q -Bernstein polynomials (see ), since the Lupas q -analogue of Bernstein operators has the singular nature at the point x = 1 and needs a new method (when x 1 , x / ( 1 - x ) ).

Remark 5.

Results similar to Theorems 1 and 2 for q -Bernstein polynomials were obtained in  and , respectively. Note that when f ( x ) = x 2 , for q ( 0,1 ) , we have (see (46)) (14) R n , q ( f ; x ) - R , q ( f ; x ) = q n x ( 1 - x ) ( 1 - x + q x ) [ n ] q n ω 2 ( f ; q n ) . Hence, the estimate (11) is sharp in the following sense: the sequence q n in (11) cannot be replaced by any other sequence decreasing to zero more rapidly as n . However, (11) is not sharp for the Lipschitz class Lip    α ( α ( 0,1 ] ) in the sense of order. This, combining with Theorem 1, shows that in the case 0 < q < 1 the modulus of continuity is more appropriate to describe the rate of convergence for the Lupas q -analogue Berstein operators than the second modulus of smoothness. This is different from that in the case q = 1 .

Remark 6.

The numbers c in (11) and C q in (9) are both the constants independent of f and n . However, while c in (11) does not depend on q , the constant C q in (9) depends on q and tends to + as q 1 - . Hence, (11) does not follow from (9).

Let f C [ 0,1 ] and g ( x ) = f ( 1 - x ) . Using (7) and the relations (15) ω ( f , t ) = ω ( g , t ) ; ω 2 ( f , t ) = ω 2 ( g , t ) , we have the following corollaries.

Corollary 7.

Let f C [ 0,1 ] . Then for any q ( 1 , ) , (16) R n , q ( f ) - R , q ( f ) C q ω ( f ; 1 q n ) , where C q is a constant independent of f and n .

Corollary 8.

Let f C [ 0,1 ] . Then for any q ( 1 , ) , (17) R n , q ( f ) - R , q ( f ) c ω 2 ( f ; 1 q n ) . Furthermore, (18) sup q > 0 | R n , q ( f ) - R , q ( f ) | c ω 2 ( f ; n - 1 / 2 ) , where c is an absolute constant.

2. Proofs of Theorems <xref ref-type="statement" rid="thm1">1</xref> and <xref ref-type="statement" rid="thm2">2</xref>

For the proofs of Theorems 1 and 2, we need the following lemmas.

Lemma 9 (see [<xref ref-type="bibr" rid="B8">2</xref>]).

The following equalities are true: (19) R n , q ( 1 ; x ) = R , q ( 1 ; x ) = 1 , R n , q ( t ; x ) = R , q ( t ; x ) = x , (20) R n , q ( t 2 ; x ) = x 2 + x ( 1 - x ) [ n ] - x 2 ( 1 - x ) ( 1 - q ) 1 - x + x q ( 1 - 1 [ n ] ) .

Lemma 10.

With the definitions of r n , k ( q ; x ) and r , k ( q ; x ) , we have (21) k = 0 n q k r n , k ( q ; x ) = 1 - x + q n x , k = 0 q k r , k ( q ; x ) = 1 - x .

Proof.

Using (19) and (3), we get (22) k = 0 n q k r n , k ( q ; x ) = ( q n - 1 ) k = 0 n q k - 1 q n - 1 r n , k ( q ; x ) + k = 0 n r n , k ( q ; x ) = ( q n - 1 ) k = 0 n [ k ] [ n ] r n , k ( q ; x ) + 1 = ( q n - 1 ) R n , q ( t ; x ) + 1 = 1 - x + q n x . Similarly, using (19) and (5), we have (23) k = 0 q k r , k ( q ; x ) = k = 0 ( q k - 1 ) r , k ( q ; x ) + k = 0 r , k ( q ; x ) = - ( k = 0 ( 1 - q k ) r , k ( q ; x ) ) + k = 0 r , k ( q ; x ) = - R , q ( t ; x ) + 1 = 1 - x . The proof of Lemma 10 is complete.

For integers n , k , and q ( 0,1 ) , x [ 0,1 ] , we have (24) r n , k ( q ; x ) - r k ( q ; x ) = [ n k ] q k ( k - 1 ) / 2 ( x / ( 1 - x ) ) k s = 0 n - 1 ( 1 + q s ( x / ( 1 - x ) ) ) - q k ( k - 1 ) / 2 ( x / ( 1 - x ) ) k ( 1 - q ) k [ k ] ! s = 0 ( 1 + q s ( x / ( 1 - x ) ) ) = [ n k ] q k ( k - 1 ) / 2 ( x / ( 1 - x ) ) k s = 0 n - 1 ( 1 + q s ( x / ( 1 - x ) ) ) × ( 1 - 1 s = n ( 1 + q s ( x / ( 1 - x ) ) ) ) + q k ( k - 1 ) / 2 ( x / ( 1 - x ) ) k s = 0 ( 1 + q s ( x / ( 1 - x ) ) ) ( [ n k ] - 1 ( 1 - q ) k [ k ] ! ) = r n , k ( q ; x ) ( 1 - 1 s = n ( 1 + q s ( x / ( 1 - x ) ) ) ) - r , k ( q ; x ) ( 1 - s = n - k + 1 n ( 1 - q s ) ) = r n , k ( q ; x ) J 1 - r , k ( q ; x ) J 2 , where (25) J 1 : = 1 - 1 s = n ( 1 + q s ( x / ( 1 - x ) ) ) , J 2 : = 1 - s = n - k + 1 n ( 1 - q s ) . We will prove the following lemma.

Lemma 11.

Let 0 < q < 1 . Then for integers n , k and for 0 < x < 1 / ( 1 + q n ) , (26) k = 0 n q k | r n , k ( q ; x ) - r , k ( q ; x ) | 3 q n 1 - q .

Proof.

It is easy to prove by induction that (27) 0 J 2 : = 1 - s = n - k + 1 n ( 1 - q s ) s = n - k + 1 n q s s = n - k q s = q n - k 1 - q . Since 1 - exp ( - x ) x and ln ( 1 + x ) x for all x [ 0 , ) , we obtain (28) 0 J 1 = 1 - exp ( - s = n ln ( 1 + q s x 1 - x ) ) s = n ln ( 1 + q s x 1 - x ) s = n q s x 1 - x = q n x ( 1 - q ) ( 1 - x ) . Hence, (29) | r n , k ( q ; x ) - r k ( q ; x ) | q n x ( 1 - q ) ( 1 - x ) r n , k ( q ; x ) + q n - k 1 - q r , k ( q ; x ) , and therefore, by (21) and (19) we get (30) k = 0 n q k | r n , k ( q ; x ) - r , k ( q ; x ) | q n x ( 1 - q ) ( 1 - x )    k = 0 n q k r n , k ( q ; x ) + q n 1 - q k = 0 n r , k ( q ; x ) q n x ( 1 - q ) ( 1 - x ) ( 1 - x + q n x ) + q n 1 - q . Since 0 < x < 1 / ( 1 + q n ) < 1 , it follows that 0 < x / ( 1 - x ) < 1 / q n and thence (31) k = 0 n q k | r n , k ( q ; x ) - r , k ( q ; x ) | 3 q n 1 - q . This completes the proof of Lemma 11.

Proof of Theorem <xref ref-type="statement" rid="thm1">1</xref>.

It follows from the definition of R n , q ( f ; x ) and R , q ( f ; x ) that both of them possess the end point interpolation property; in other words, (32) R n , q ( f ; 0 ) = R , q ( f ; 0 ) = f ( 0 ) , R n , q ( f ; 1 ) = R , q ( f ; 1 ) = f ( 1 ) . It follows from the definition of r n , k ( q ; x ) and r , k ( q ; x ) that r n , k ( q ; x ) 0 and r , k ( q ; x ) 0 for 0 x < 1 . If x 1 , then x / ( 1 - x ) . So, the Lupas q -analogue of Bernstein operators has the singular nature at the point x = 1 and the rate of convergence near the point 1 needs to be considered independently. First we suppose x ( 1 / ( 1 + q n ) , 1 ) ; that is, 1 - x < q n / ( 1 + q n ) < q n . Then (33) I = | R n , q ( f ; x ) - R , q ( f ; x ) | = | k = 0 n ( f ( [ k ] [ n ] ) - f ( 1 ) ) r n , k ( q ; x ) h h h h h - k = 0 ( f ( 1 - q k ) - f ( 1 ) ) r , k ( q ; x ) | k = 0 n | f ( [ k ] [ n ] ) - f ( 1 ) | r n , k ( q ; x ) + k = 0 | f ( 1 - q k ) - f ( 1 ) | r , k ( q ; x ) . Since (34) | [ k ] [ n ] - 1 | = | 1 - q k 1 - q n - 1 | q k ( 1 - q n - k ) 1 - q n q k , ( 0 k n ) , ω ( f ; λ t ) ( 1 + λ ) ω ( f ; t ) , λ > 0 , we get (35) I k = 0 n ω ( f ; q k ) r n , k ( q ; x ) + k = 0 ω ( f ; q k ) r , k ( q ; x ) k = 0 n ω ( f , q n ) ( 1 + q k q n ) r n , k ( q ; x ) + k = 0 ω ( f ; q n ) ( 1 + q k q n ) r , k ( q ; x ) 2 ω ( f ; q n ) + ω ( f , q n ) q n k = 0 n q k r n , k ( q ; x ) + ω ( f , q n ) q n k = 0 q k r , k ( q ; x ) . By Lemma 10 and 1 - x < q n , x < 1 , we have (36) I 2 ω ( f ; q n ) + ω ( f , q n ) q n ( 1 - x + q n x ) + ω ( f , q n ) q n ( 1 - x ) 5 ω ( f ; q n ) .

Next, we assume that 0 < x < 1 / ( 1 + q n ) . Then 0 x / ( 1 - x ) 1 / q n . We have (37) I = | R n , q ( f ; x ) - R , q ( f ; x ) | = | k = 0 n ( f ( [ k ] [ n ] ) - f ( 1 - q k ) ) r n , k ( q ; x ) hhhh + k = 0 n ( f ( 1 - q k ) - f ( 1 ) ) ( r n , k ( q ; x ) - r , k ( q ; x ) ) hhhh - k = n + 1 ( f ( 1 - q k ) - f ( 1 ) ) r , k ( q ; x ) | k = 0 n | f ( [ k ] [ n ] ) - f ( 1 - q k ) | r n , k ( q ; x ) + k = 0 n | f ( 1 - q k ) - f ( 1 ) | | r n , k ( q ; x ) - r , k ( q ; x ) | + k = n + 1 | f ( 1 - q k ) - f ( 1 ) | r , k ( q ; x ) = : δ 1 + δ 2 + δ 3 . First we estimate δ 1 and δ 3 . Since (38) | [ k ] [ n ] - ( 1 - q k ) | = | 1 - q k 1 - q n - ( 1 - q k ) | = q n ( 1 - q k ) 1 - q n q n , ( 0 k n ) | 1 - ( 1 - q k ) | = q k q n , ( k n + 1 ) , we get (39) δ 1 ω ( f , q n ) k = 0 n r n , k ( q ; x ) = ω ( f , q n ) , (40) δ 3 ω ( f , q n ) k = n + 1 r , k ( q ; x ) ω ( f , q n ) . Now we estimate δ 2 . Since ω ( f , λ t ) ( 1 + λ ) ω ( f , t ) , by Lemma 11 we get (41) δ 2 k = 0 n ω ( f , q k ) | r n , k ( q ; x ) - r , k ( q ; x ) | k = 0 n ω ( f , q n ) ( 1 + q k q n ) | r n , k ( q ; x ) - r , k ( q ; x ) | 2 ω ( f ; q n ) q n k = 0 n q k | r n , k ( q ; x ) - r , k ( q ; x ) | 6 ω ( f ; q n ) 1 - q . From (39)–(41), we have for 0 x 1 / ( 1 + q n ) , (42) I ( 2 + 6 1 - q ) ω ( f ; q n ) . Hence from (36) and (42), we conclude that, for q ( 0,1 ) , (43) R n , q ( f ; x ) - R , q ( f ; x ) C q    ω ( f ; q n ) , where C q = 2 + 6 / ( 1 - q ) .

At last we show that the estimate (9) is sharp. For each α , 0 < α 1 , suppose that f α ( x ) is a continuous function, which is equal to zero in [ 0,1 - q ] and [ 1 - q 2 , 1 ] , equal to ( x - ( 1 - q ) ) α in [ 1 - q , 1 - q + q ( 1 - q ) / 2 ] , and linear in the rest of [ 0,1 ] . It is obvious that ω ( f α ; t ) c t α , and (44) R n , q ( f α ) - R , q ( f α ) q n α | r n , 1 ( q ; · ) | q n α . The proof of Theorem 1 is complete.

In order to prove Theorem 2, we need the following result.

Theorem A (see [<xref ref-type="bibr" rid="B19">12</xref>]).

Let the sequence { L n } of positive linear operators on C [ 0,1 ] satisfy the following conditions.

The sequence { L n ( e 2 ) } converges to a function L ( e 2 ) in C [ 0,1 ] , where e i ( x ) = x i , i = 0,1 , 2 .

The sequence { L n ( f , x ) } n 1 is nonincreasing for any convex function f and for any x [ 0,1 ] .

Then there exists an operator L on C [ 0,1 ] such that L n ( f ) - L ( f ) 0 for any f C [ 0,1 ] . Furthermore, (45) | L n ( f , x ) - L ( f , x ) | c ω 2 ( f ; λ n ( x ) ) , where λ n ( x ) = L n ( e 2 , x ) - L ( e 2 , x ) and c is a constant which depends only on L 1 ( e 0 ) .

Proof of Theorem <xref ref-type="statement" rid="thm2">2</xref>.

From , we know that the Lupas q -analogues of the Bernstein operators satisfy Condition (B). It follows from  that, for q ( 0,1 ) , { R n , q ( f ; x ) } converges to R , q ( f ; x ) uniformly in x [ 0,1 ] as n ; and (46) 0 λ n ( x ) = R n , q ( t 2 , x ) - R , q ( t 2 , x ) = R n , q ( t 2 , x ) - lim n R n , q ( t 2 ; x ) = x ( 1 - x ) [ n ] - x 2 ( 1 - x ) ( 1 - q ) 1 - x + x q ( 1 - 1 [ n ] ) - x ( 1 - x ) ( 1 - q ) + x 2 ( 1 - x ) ( 1 - q ) q 1 - x + x q = x ( 1 - x ) ( 1 [ n ] - ( 1 - q ) ) + x 2 ( 1 - x ) ( 1 - q ) 1 - x + x q ( 1 [ n ] - ( 1 - q ) ) = x ( 1 - x ) 1 - x + x q ( 1 - q ) q n 1 - q n q n . Theorem 2 follows from (46) and Theorem A.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Heping Wang is supported by the National Natural Science Foundation of China (Project no. 11271263), the Beijing Natural Science Foundation (1132001), and BCMIIS. Yanbo Zhang is supported by the Science and Technology Plan Projects of Higher Institutions in Shandong Province (J13LI55).

Bernstein S. N. Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités Communications of the Mathematical Society of Charkow Séries 2 1912 13 1 2 Lupas A. A q -analogue of the Bernstein operator University of Cluj-Napoca seminar on numerical and statistical calculus, 9, 1987 Phillips G. M. Bernstein polynomials based on the q -integers Annals of Numerical Mathematics 1997 4 1–4 511 518 MR1422700 Lewanowicz S. Woźny P. Generalized Bernstein polynomials BIT Numerical Mathematics 2004 44 1 63 78 2-s2.0-4043127644 10.1023/B:BITN.0000025086.89121.d8 MR2057362 ZBL1045.41003 Derriennic M.-M. Modified Bernstein polynomials and Jacobi polynomials in q -calculus Rendiconti del Circolo Matematico di Palermo Serie 2 2005 supplement 76 269 290 MR2178441 Il'inskii A. Ostrovska S. Convergence of generalized Bernstein polynomials Journal of Approximation Theory 2002 116 1 100 112 2-s2.0-0036287075 10.1006/jath.2001.3657 MR1909014 Ostrovska S. On the q -Bernstein polynomials Advanced Studies in Contemporary Mathematics 2005 11 2 193 204 MR2169894 Ostrovska S. The approximation by q -Bernstein polynomials in the case q 1 Archiv der Mathematik 2006 86 3 282 288 2-s2.0-33644961254 10.1007/s00013-005-1503-y MR2215317 Ostrovska S. The first decade of the q -Bernstein polynomials: results and perspectives Journal of Mathematical Analysis and Approximation Theory 2007 2 1 35 51 MR2477320 Phillips G. M. Interpolation and Approximation by Polynomials 2003 New York, NY, USA Springer MR1975918 Phillips G. M. A survey of results on the q -Bernstein polynomials IMA Journal of Numerical Analysis 2010 30 1 277 288 2-s2.0-76549126480 10.1093/imanum/drn088 MR2580559 Wang H. Korovkin-type theorem and application Journal of Approximation Theory 2005 132 2 258 264 2-s2.0-13344287024 10.1016/j.jat.2004.12.010 MR2118520 ZBL1118.41015 Wang H. Voronovskaya-type formulas and saturation of convergence for q -Bernstein polynomials for 0 < q < 1 Journal of Approximation Theory 2007 145 2 182 195 2-s2.0-33947286902 10.1016/j.jat.2006.08.005 MR2312464 Wang H. Meng F. The rate of convergence of q -Bernstein polynomials for 0 < q < 1 Journal of Approximation Theory 2005 136 2 151 158 2-s2.0-26844535032 10.1016/j.jat.2005.07.001 MR2171684 Wang H. Wu X. Saturation of convergence for q -Bernstein polynomials in the case q 1 Journal of Mathematical Analysis and Applications 2008 337 1 744 750 2-s2.0-34548252925 10.1016/j.jmaa.2007.04.014 MR2356108 Finta Z. Quantitative estimates for the Lupaş q -analogue of the Bernstein operator Demonstratio Mathematica 2011 44 1 123 130 2-s2.0-79961094525 MR2796767 Han L.-W. Chu Y. Qiu Z.-Y. Generalized Bézier curves and surfaces based on Lupaş q -analogue of Bernstein operator Journal of Computational and Applied Mathematics 2014 261 352 363 10.1016/j.cam.2013.11.016 MR3144717 Mahmudov N. I. Sabancigil P. Voronovskaja type theorem for the Lupaş q -analogue of the Bernstein operators Mathematical Communications 2012 17 1 83 91 MR2946134 Ostrovska S. On the Lupas q -analogue of the Bernstein operator The Rocky Mountain Journal of Mathematics 2006 36 5 1615 1629 2-s2.0-33846826629 10.1216/rmjm/1181069386 MR2285304 Ostrovska S. On the Lupaş q -transform Computers & Mathematics with Applications 2011 61 3 527 532 2-s2.0-78951472266 10.1016/j.camwa.2010.11.025 MR2764046 Ostrovska S. Analytical properties of the Lupaş q -transform Journal of Mathematical Analysis and Applications 2012 394 1 177 185 10.1016/j.jmaa.2012.04.047 MR2926214 Lorentz G. G. Bernstein Polynomials 1986 New York, NY, USA Chelsea MR864976