AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 259491 10.1155/2014/259491 259491 Research Article Approximation by q -Bernstein Polynomials in the Case q 1 + Wu Xuezhi Ostrovska Sofiya Library Capital Normal University Beijing 100048 China cnu.edu.cn 2014 1132014 2014 24 12 2013 08 02 2014 12 3 2014 2014 Copyright © 2014 Xuezhi Wu. 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.

Let B n , q ( f ; x ) , q ( 0 , ) be the q -Bernstein polynomials of a function f C [ 0,1 ] . It has been known that, in general, the sequence B n , q n ( f ) with q n 1 + is not an approximating sequence for f C [ 0,1 ] , in contrast to the standard case q n 1 - . In this paper, we give the sufficient and necessary condition under which the sequence B n , q n ( f ) approximates f for any f C [ 0,1 ] in the case q n > 1 . Based on this condition, we get that if 1 < q n < 1 + ln 2 / n for sufficiently large n , then B n , q n ( f ) approximates f for any f C [ 0,1 ] . On the other hand, if B n , q n ( f ) can approximate f for any f C [ 0,1 ] in the case q n > 1 , then the sequence ( q n ) satisfies lim ¯ n n ( q n - 1 ) ln 2 .

1. Introduction

Let q > 0 . For any nonnegative integer k , the q -integer   [ k ] q is defined by (1) [ k ] q : = 1 + q + + q k - 1 , ( k = 1,2 , ) , [ 0 ] q : = 0 , and the q -factorial   [ k ] q ! by (2) [ k ] q ! [ 1 ] q [ 2 ] q [ k ] q , ( k = 1,2 , ) , [ 0 ] q ! : = 1 . For integers k , n with 0 k n , the q -binomial coefficient is defined by (3) [ n k ] q : = [ n ] q ! [ k ] q ! [ n - k ] q ! . In , Phillips proposed the q -Bernstein polynomials: for each positive integer n and f C [ 0,1 ] , the q -Bernstein polynomial of f is (4) B n , q ( f ; x ) : = k = 0 n f ( [ k ] q [ n ] q ) p n k ( q ; x ) , where (5) p n k ( q ; x ) = [ n k ] q x k s = 0 n - k - 1 ( 1 - q s x ) . Note that, for q = 1 , B n , q ( f ; x ) is the classical Bernstein polynomial B n ( f ; x ) : (6) B n ( f ; x ) : = k = 0 n f ( k n ) ( n k ) x k ( 1 - x ) n - k .

In recent years, the q -Bernstein polynomials have been investigated intensively and a great number of interesting results related to the q -Bernstein polynomials have been obtained. Reviews of the results on q -Bernstein polynomials are given in [2, Chapter 7] and [3, 4].

The q -Bernstein polynomials inherit some of the properties of the classical Bernstein polynomials, for example, the end-point interpolation property and the shape-preserving properties in the case 0 < q < 1 , representation via divided differences. We can also define the generalized Bézier curve and de Casteljau algorithm, which can be used for evaluating q -Bernstein polynomials iteratively. These properties stipulate the importance of q -Bernstein polynomials for the computer-aided geometric design. Like the classical Bernstein polynomials, the q -Bernstein polynomials reproduce linear functions and are degree reducing on the set of polynomials. Apart from that, the basic q -Bernstein polynomials p n k ( q ; x ) admit a probabilistic interpretation via the stochastic process and the q -binomial distribution in the case 0 < q < 1 ; see .

On the other hand, when passing from q = 1 to q 1 convergence properties of the q -Bernstein polynomials dramatically change. More specially, in the case 0 < q < 1 , B n , q are positive linear operators on C [ 0,1 ] , and the convergence properties of the q -Bernstein polynomials have been investigated intensively (see, e.g., ). In the case q > 1 , B n , q are not positive linear operators on C [ 0,1 ] , and the lack of positivity makes the investigation of convergence in the case q > 1 essentially more difficult. There are many unexpected results concerning convergence of q -Bernstein polynomials in the case q > 1 (see [2, 1217]). For example, the rate of approximation by q -Bernstein polynomials ( q > 1 ) in C [ 0,1 ] for functions analytic in { z : | z | < q + ɛ } is q - n versus 1 / n for the classical Bernstein polynomials, while, for some infinitely differentiable functions on [ 0,1 ] , their sequences of q -Bernstein polynomials ( q > 1 ) may be divergent (see ). In [2, 15], strong asymptotic estimates for the norm B n , q as n for fixed q > 1 and as q are obtained. It was shown in  that B n , q + faster than any geometric progression n for fixed q > 1 . This fact provides an explanation for the unpredictable behavior of q -Bernstein polynomials ( q > 1 ) with respect to convergence.

This paper is devoted to studying approximation properties of q -Bernstein polynomials for q taking varying values that tend to 1. We note that, from the very first papers (see ), there was interest in such approximation properties. In the case 0 < q n < 1 , many interesting results including the convergence, the rate of convergence, Voronvskaya-type theorems, and the direct and converse theorem are obtained (see [1, 6, 811]). It was shown in [1, 8] that, in the case q n 1 , the condition q n 1 is necessary and sufficient for the sequence ( B n , q n ( f ) ) to be approximating for any f C [ 0,1 ] .

Naturally, the question arises as to whether the sequence ( B n , q n ( f ) ) to be approximating for any f C [ 0,1 ] as q n tends to 1 from above. It turns out that, in general, the answer is negative. Indeed, Ostrovska showed in  that if q n - 1 0 slower than ( ln n ) / n , then the sequence ( B n , q n ( f ) ) may not be approximating for some f C [ 0,1 ] (e.g., f ( x ) = x ). However, in  Ostrovska showed that if q n 1 + fast enough, the sequence ( B n , q n ( f ) ) is approximating for any f C [ 0,1 ] : a sufficient condition is q n = 1 + o ( n - 1 3 - n ) .

In this paper, we continue to study the convergence of the sequence ( B n , q n ) as q n tends to 1 from above. Clearly, the convergence of the sequence ( B n , q n ) depends heavily on the operator norms B n , q . We remark that for B n , q n = 1 for all 0 < q n < 1 . In contrast to this, B n , q n vary with q n > 1 . By the delicate analysis of B n , q n , we obtain the sufficient and necessary condition under which ( B n , q n ( f ; · ) )    ( q n > 1 ) approximates f for any f C [ 0,1 ] . Based on this condition we get that if ( B n , q n ( f ; · ) ) can approximate f for any f C [ 0,1 ] , then the sequence ( q n ) satisfies lim ¯ n n ( q n - 1 ) ln 2 . On the other hand, if 1 < q n 1 + ln 2 / n for sufficient large n , then ( B n , q n ( f ; · ) ) approximates f for any f C [ 0,1 ] .

2. Statement of Results

From here on we assume that q n > 1 . The following theorem gives the sufficient and necessary condition for convergence of the sequence ( B n , q n ( f ) ) for any f C [ 0,1 ] .

Theorem 1.

Let q n > 1 . Then the sequence ( B n , q n ( f ) ) converges to f in C [ 0,1 ] for any f C [ 0,1 ] if and only if (7) sup n sup x [ q n - 1 , 1 ] k = 2 n | p n    n - k ( q n ; x ) | < .

Based on Theorem 1, we obtain the following necessary condition for convergence of the sequence ( B n , q n ( f ) ) . Indeed, we show that if lim ¯ n n ( q n - 1 ) > ln 2 , then sup n | p n    n - [ ln n ] ( q n ; x 0 ) | = with x 0 = ( 1 + q n ) / 2 q n .

Theorem 2.

Let q n > 1 . If the sequence ( B n , q n ( f ) ) converges to f in C [ 0,1 ] , for any f C [ 0,1 ] , then (8) lim n ¯ n ( q n - 1 ) ln 2 .

Finally, we give the sufficient condition for convergence of the sequence ( B n , q n ( f ) ) .

Theorem 3.

Let q n > 1 . If the sequence ( q n ) satisfies q n 1 + ln 2 / n for sufficiently large n , then, for any f C [ 0,1 ] , ( B n , q n ( f ; x ) ) converges to f ( x ) uniformly on [ 0,1 ] .

The following corollary follows immediately for Theorem 3.

Corollary 4.

Let q n > 1 . If the sequence ( q n ) satisfies (9) lim n ¯ n ( q n - 1 ) < ln 2 , then, for any f C [ 0,1 ] , ( B n , q n ( f ; x ) ) converges to f ( x ) uniformly on [ 0,1 ] .

Remark 5.

Using the same technique as in the proof of Theorem 3, we can prove a slightly stronger conclusion: if (10) 1 < q n 1 + ln 2 n + C n 2 for some positive constant C and sufficiently large n , then, for any f C [ 0,1 ] , ( B n , q n ( f ; x ) ) converges to f ( x ) uniformly on [ 0,1 ] .

3. Proofs of Theorems <xref ref-type="statement" rid="thm1">1</xref>–<xref ref-type="statement" rid="thm3">3</xref>

For f C [ 0,1 ] , we set (11) f : = max x [ 0,1 ] | f ( x ) | , f s : = f C [ q n - s - 1 , q n - s ] : = max x [ q n - s - 1 , q n - s ] | f ( x ) | . Let F n ( x ) : = k = 0 n | p n k ( q n ; x ) | , x [ 0,1 ] . Clearly, (12) B n , q n = F n = max x [ 0,1 ] ( k = 0 n | p n    n - k ( q n ; x ) | ) . Note that k = 0 n p n k ( q n ; x ) = 1 for x [ 0,1 ] and p n k ( q n ; x ) 0 for x [ 0 , q n - n + 1 ] and k = 0,1 , , n . This means that (13) F n ( x ) = 1 , x [ 0 , q n - n + 1 ] , F n ( x ) 1 , x [ q n - n + 1 , 1 ] . It follows that (14) B n , q n = F n = max 0 s n - 2 F n s .

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

From Corollary 7 in  we know that, for any polynomial P ( x ) , we have (15) B n , q n ( P ; x ) P ( x ) uniformly in [ 0,1 ] as n . It follows from the well-known Banach-Steinhaus theorem that ( B n , q n ( f ) ) ( q n > 1 ) approximates f for any f C [ 0,1 ] if and only if (16) sup n B n , q n = sup n sup x [ 0,1 ] ( k = 0 n | p n    n - k ( q n ; x ) | ) < + . We set (17) G s , n ( x ) = k = s + 2 n | p n    n - k ( q n ; x ) | , s = 0,1 , , n - 2 . Since p n    n - k ( q n ; x ) 0 for x [ q n - s - 1 , q n - s ] and k = 0,1 , , s + 1 , we get, for x [ q n - s - 1 , q n - s ] , (18) k = 0 s + 1 | p n    n - k ( q n ; x ) | = k = 0 s + 1 p n    n - k ( q n ; x ) = 1 - k = s + 2 n p n    n - k ( q n ; x ) 1 + G s , n ( x ) , and, therefore, (19) F n s 1 + 2 G s , n s = 1 + 2 G s , n s , s = 0,1 , , n - 2 . Next we will show that (20) G s , n s G s - 1 , n s - 1 , s = 1,2 , , n - 2 . Note that, for x [ q n - s - 1 , q n - s ] , (21) G s , n ( x ) = k = s + 2 n | p n    n - k ( q n ; x ) | , G s - 1 , n ( q n x ) = k = s + 1 n | p n    n - k ( q n ; q n x ) | . If we show that, for x [ q n - s - 1 , q n - s ] and k = s + 1 , , n - 1 , (22) | p n    n - k - 1 ( q n ; x ) | | p n    n - k ( q n ; q n x ) | , then (23) G s , n ( x ) G s - 1 , n ( q n x ) , x [ q n - s - 1 , q n - s ] , and (20) follows. Indeed, for x ( q n - s - 1 , q n - s ) and k = s + 1 , , n - 1 , (24) | p n    n - k ( q n ; q n x ) | | p n    n - k - 1 ( q n ; x ) | = [ n k ] q n ( q n x ) n - k j = 0 s - 1 ( 1 - q n j + 1 x ) j = s k - 1 ( q n j + 1 x + 1 ) [ n k + 1 ] q n x n - k - 1 j = 0 s ( 1 - q n j x ) j = s + 1 k ( q n j x - 1 ) = [ k + 1 ] q n q n n - k x [ n - k ] q n ( 1 - x ) . Hence, (22) is equivalent to the following inequality: (25) ( q n k + 1 - 1 ) q n n - k x ( q n n - k - 1 ) ( 1 - x ) , which is also equivalent to the inequality (26) x q n n - k - 1 q n n + 1 - 1 . For x ( q n - s - 1 , q n - s ) and k = s + 1 , , n - 1 , we have (27) x > q n - s - 1 q n - s - 1 ( q n n - q n s + 1 ) q n n + 1 - 1 = q n n - s - 1 - 1 q n n + 1 - 1 q n n - k - 1 q n n + 1 - 1 . This proves (26). On the other hand, p n    n - k - 1 ( q n ; x ) = 0 = p n    n - k ( q n ; q n x ) for x { q n - s - 1 , q n - s } , which completes the proof of (20). From (14), (19), and (20), we get (28) G 0 , n 0 F n = B n , q n 1 + 2 G 0 , n 0 . This implies that (16) is equivalent to (29) sup n G 0 , n 0 = sup n sup x [ q n - 1 , 1 ] k = 2 n | p n    n - k ( q n ; x ) | < . Theorem 1 is proved.

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

First we show that (30) q n - 1 = O ( 1 n ) . Otherwise, we may assume that (31) lim n n ( q n - 1 ) = + , which implies (32) lim n q n n - 1 = lim n exp ( ( n - 1 ) ln q n ) lim n exp ( ( n - 1 ) min { ( q n - 1 ) 2 , ln 2 } ) = + . We have (33) G 0 , n 0 p n    n - 2 ( q n ; · ) 0 | p n    n - 2 ( q n ; q n + 1 2 q n ) | = ( q n n - 1 ) ( q n n - 1 - 1 ) ( q n 2 - 1 ) ( q n - 1 ) ( 1 + q n 2 q n ) n - 2 × ( 1 - 1 + q n 2 q n ) ( q n 1 + q n 2 q n - 1 ) = ( 1 - q n - n + 1 ) ( q n n - 1 ) 8 ( 1 + q n 2 ) n - 3 ( 1 - q n - n + 1 ) ( q n n - 1 ) 8 + , ( as n ) . This leads to a contradiction by Theorem 1. Hence, (30) holds.

Next, we show Theorem 2. Assume that lim ¯ n n ( q n - 1 ) > ln 2 . Then by (30) we may suppose that, for some A , B , ln 2 < A < B < + , (34) 1 + A n q n 1 + B n .

For 0 < a < b , we set h ( x ) = ( x a - 1 ) / ( x b - 1 ) , x > 1 . Direct computation gives that (35) h ( x ) = b x a - 1 ( x b - a - ( ( b - a ) / b ) x b - a / b ) ( x b - 1 ) 2 . Since the function g ( y ) = x y is convex on ( - , + ) for a fixed x > 0 , we get that (36) x b - a = x ( ( b - a ) / b ) · b + ( a / b ) · 0 b - a b x b + a b . This means that h ( x ) 0 and h ( x ) is nonincreasing on ( 1 , + ) . Hence, for x ( 1 , ξ 0 ) , ξ 0 > 1 , we have (37) h ( ξ 0 ) h ( x ) lim x 1 + h ( x ) = a b . Put x 0 = ( 1 + q n ) / 2 q n ( q n - 1 , 1 ) . Then, for k 0 = [ ln n ] , we have (38) G 0 , n 0 p n    n - k 0 ( q n ; · ) 0 | p n    n - k 0 ( q n ; x 0 ) | = ( q n n - 1 ) ( q n n - k 0 + 1 - 1 ) ( q n k 0 - 1 ) ( q n - 1 ) x 0 n - k 0 × ( 1 - x 0 ) s = 1 k 0 - 1 ( q n s x 0 - 1 ) ( q n n - k 0 - 1 ) k 0 x 0 n - k 0 ( 1 - x 0 ) × ( q n k 0 - 1 x 0 - 1 ) ( q n x 0 - 1 ) ( q n k 0 - 1 ) ( q n - 1 ) . Using (34), the inequalities (39) q n s + 1 x 0 - 1 q n s - 1 1 , s = 1 , , k 0 - 2 , x 0 n - k 0 ( 1 - x 0 ) ( q n x 0 - 1 ) q n - n + k 0 - 1 ( q n - 1 ) 2 4 ( 1 + B n ) - n ( q n - 1 ) 2 4 ( q n - 1 ) 2 exp ( - B ) 4 , and the nonincreasing property of h ( x ) , we continue to obtain that (40) G 0 , n 0 ( ( 1 + A n ) n - ln n - 1 ) k 0 exp ( - B ) 4 ( q n - 1 ) 2 ( q n k 0 - 1 ) ( q n k 0 - 1 - 1 ) ( ( 1 + A n ) n - ln n - 1 ) k 0 × exp ( - B ) 4 ( A / n ) 2 ( ( 1 + B / n ) k 0 - 1 ) ( ( 1 + B / n ) k 0 - 1 - 1 ) . We observe that (41) lim n ( 1 + A n ) n - ln n = exp ( lim n ( n - ln n ) ln ( 1 + A n ) ) = exp ( lim n A ( n - ln n ) n ) = exp ( A ) > 2 , and, for s = k 0 , k 0 - 1 , (42) lim n ( 1 + B / n ) s - 1 B ln n / n = lim n exp ( s ln ( 1 + B / n ) ) - 1 B ln n / n = lim n s ln ( 1 + B / n ) B ln n / n = 1 . Thus, for some a ( 1 , e A - 1 ) and sufficiently large n , we have (43) G 0 , n 0 a ln n - 1 ( ln n ) 2 exp ( - B ) A 2 4 B 2 + . By Theorem 1, we know that there exists a function f C [ 0,1 ] such that the sequence ( B n , q n ( f ) ) does not converge to f in C [ 0,1 ] . This leads to a contradiction. Hence, lim ¯ n n ( q n - 1 ) ln 2 . Theorem 2 is proved.

Proof of Theorem <xref ref-type="statement" rid="thm3">3</xref>.

From Theorem 1, we know that it is sufficient to show that if q n 1 + ln 2 / n for sufficiently large n , then (44) sup n G 0 , n 0 < . For x ( q n - 1 , 1 ) , we set α = - log q n x . Then α ( 0,1 ) and x = q n - α . Since, for k = 2 , , n - 1 , (45) q n α ( q n n - k - 1 ) q n n - k + α - 1 q n n - 1 ( 1 + ln 2 n ) n - 1 1 , by (37) we get that (46) | p n    n - k - 1 ( q n ; x ) | | p n    n - k ( q n ; x ) | = ( q n n - k - 1 ) ( q n k - α - 1 ) ( q n k + 1 - 1 ) q n - α q n k - α - 1 q n k + 1 - 1 k - α k + 1 . On the other hand, by (37) we have (47) | p n    n - 2 ( q n ; x ) | = [ n 2 ] q n x n - 1 ( 1 x - 1 ) ( q n x - 1 ) ( q n n - 1 ) ( q n n - 1 - 1 ) ( q n 2 - 1 ) ( q n - 1 ) ( q n α - 1 ) ( q n 1 - α - 1 ) ( q n α - 1 ) ( q n 1 - α - 1 ) 2 ( q n - 1 ) 2 α ( 1 - α ) 2 . It follows from (46) and (47) that (48) | p n    n - k ( q n ; x ) | α ( 1 - α ) ( k - 1 - α ) k ! . Hence, for x = q n - α , α ( 0,1 ) , (49) G 0 , n ( x ) = k = 2 n | p n    n - k ( q n ; x ) | k = 2 α ( 1 - α ) ( k - 1 - α ) k ! . Obviously (49) is satisfied for x { 0,1 } . We note that, for x [ 0,1 ] , (50) ( 1 - x ) α = 1 - α x - k = 2 α ( 1 - α ) ( k - 1 - α ) k ! x k . The above formula with x = 1 means that (51) k = 2 α ( 1 - α ) ( k - 1 - α ) k ! = 1 - α . Thus, by (49), (52) G 0 , n 0 sup α [ 0,1 ] k = 2 α ( 1 - α ) ( k - 1 - α ) k ! = 1 . This completes the proof of Theorem 3.

Conflict of Interests

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

Acknowledgments

The author is very grateful to S. Ostrovska and the anonymous referees for their useful comments and suggestions that helped improve the presentation of the paper. The research is supported by the National Natural Science Foundation of China (Project no. 11271263) and the Beijing Natural Science Foundation (1132001).

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