The Rate of Convergence of Lupas 𝑞-Analogue of the Bernstein Operators

Abstract

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 [1]) 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 [2]). 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 [3]), the two-parametric generalization of 𝑞-Bernstein polynomials (see [4]), and the 𝑞-Bernstein-Durrmeyer operator (see [5]), 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, 6–15]). The Lupas 𝑞-analogues of the Bernstein operators {𝑅_𝑛,𝑞(𝑓;𝑥)} are less known; see [2, 16–21]. 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)≤𝑐𝐵(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 [𝑘]∶=[𝑘]_𝑞∶=1−𝑞^𝑘[𝑘][𝑘]⋯[1]1−𝑞,𝑞≠1𝑘,𝑞=1,!∶=][𝑘−1,𝑘≥11,𝑘=0. (1) For integers 0≤𝑘≤𝑛, the 𝑞-binomial coefficient is defined by 𝑛𝑘[𝑛]!∶=[𝑘]![]!.𝑛−𝑘 (2)

In [2], Lupas proposed the 𝑞-analogue of the Bernstein operator 𝑅_𝑛,𝑞(𝑓;𝑥): for each positive integer 𝑛, and 𝑓∈𝐶[0,1], 𝑅_𝑛,𝑞𝑓,𝑥∶=^𝑛_𝑘=0𝑓[𝑘][𝑛]𝑟_𝑛,𝑘𝑓(1)𝑞,𝑥,0≤𝑥<1,𝑥=1, (3) where 𝑟_𝑛,𝑘𝑛𝑘𝑞𝑞;𝑥∶=^{𝑘(𝑘−1)/2}𝑥^𝑘(1−𝑥)^𝑛−𝑘1−𝑥+𝑞𝑥⋯1−𝑥+𝑞^𝑛−1𝑥=𝑛𝑘𝑞^{𝑘(𝑘−1)/2}(𝑥/(1−𝑥))^𝑘∏^𝑛−1_𝑗=01+𝑞^𝑗((.𝑥/1−𝑥)) (4)

In [19], 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 𝑅_∞,𝑞𝑓,𝑥=^∞_𝑘=0𝑓1−𝑞^𝑘𝑟_∞𝑘𝑓(1)𝑞;𝑥,0≤𝑥<1,𝑥=1,𝑟_∞,𝑘𝑞𝑞;𝑥∶=^{𝑘(𝑘−1)/2}(𝑥/(1−𝑥))^𝑘1−𝑞^𝑘[𝑘]!∏^∞_𝑗=01+𝑞^𝑗((.𝑥/1−𝑥)) (5)(6) When 𝑞>1, the following relations (see [19]) allow us to reduce to the case 𝑞∈(0,1): 𝑅_𝑛,𝑞𝑓;𝑥=𝑅_𝑛,1/𝑞,𝑅𝑔;1−𝑥_∞,𝑞𝑓;𝑥=𝑅_∞,1/𝑞𝑔;1−𝑥, (7) 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 𝜔₂(𝑓,𝑡) are defined as follows: 𝜔𝑓;𝑡∶=sup_{[]𝑥−𝑦≤𝑡𝑥,𝑦∈0,1}𝑓(𝑥)𝑦;𝜔−𝑓₂𝑓,𝑡∶=sup_0<ℎ≤𝑡sup_{𝑥∈[0,1−2ℎ]}𝑓(𝑥+2ℎ)−2𝑓(𝑥+ℎ)+𝑓(𝑥). (8)

The main results of the paper are as follows.

Theorem 1. Let 𝑞∈(0,1) and let 𝑓∈𝐶[0,1]. Then 𝑅_𝑛,𝑞𝑓−𝑅_∞,𝑞𝑓≤𝐶_𝑞𝜔𝑓;𝑞^𝑛, (9) where𝐶_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 class𝐿𝑖𝑝𝛼∶={𝑓∈𝐶[0,1]∣𝜔(𝑓;𝑡)≪𝑡^𝛼} such that 𝑅_𝑛,𝑞𝑓_𝛼−𝑅_∞,𝑞𝑓_𝛼≍𝑞^𝑛𝛼. (10)

Theorem 2. Let0<𝑞<1. Then 𝑅_𝑛,𝑞𝑓−𝑅_∞,𝑞𝑓≤𝑐𝜔₂𝑓;𝑞^𝑛. (11) Furthermore, sup_0<𝑞<1𝑅_𝑛,𝑞𝑓−𝑅_∞,𝑞𝑓≤𝑐𝜔₂𝑓;𝑛^−1/2, (12) where 𝑐 is an absolute constant.

Remark 3. From (12), it follows that, for each 𝑓∈𝐶[0,1], lim_𝑛→∞𝑅_𝑛,𝑞𝑓;𝑥=𝑅_∞,𝑞𝑓;𝑥 (13) uniformly not only in 𝑥∈[0,1], and but also in 𝑞∈(0,1], which generalizes the Ostrovska's result in [19].

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 [22]). 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 [14]), 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 [14] and [12], respectively. Note that when 𝑓(𝑥)=𝑥², for 𝑞∈(0,1), we have (see (46)) 𝑅_𝑛,𝑞𝑓;𝑥−𝑅_∞,𝑞=𝑞𝑓;𝑥_𝑛𝑥(1−𝑥)1−𝑥+𝑞𝑥[𝑛]≍𝑞^𝑛≍𝜔₂𝑓;𝑞^𝑛. (14) 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 𝐿𝑖𝑝𝛼(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 𝜔𝑓,𝑡=𝜔𝑔,𝑡;𝜔₂𝑓,𝑡=𝜔₂𝑔,𝑡, (15) we have the following corollaries.

Corollary 7. Let 𝑓∈𝐶[0,1]. Then for any 𝑞∈(1,∞), 𝑅_𝑛,𝑞𝑓−𝑅_∞,𝑞𝑓≤𝐶_𝑞𝜔1𝑓;𝑞^𝑛, (16) where𝐶_𝑞 is a constant independent of 𝑓 and 𝑛.

Corollary 8. Let 𝑓∈𝐶[0,1]. Then for any 𝑞∈(1,∞), 𝑅_𝑛,𝑞𝑓−𝑅_∞,𝑞𝑓≤𝑐𝜔₂𝑓;1𝑞^𝑛. (17) Furthermore, sup_𝑞>0𝑅_𝑛,𝑞𝑓−𝑅_∞,𝑞𝑓≤𝑐𝜔₂𝑓;𝑛^−1/2, (18) where 𝑐 is an absolute constant.

2. Proofs of Theorems 1 and 2

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

Lemma 9 (see [2]). The following equalities are true: 𝑅_𝑛,𝑞(1;𝑥)=𝑅_∞,𝑞()𝑅1;𝑥=1,_𝑛,𝑞(𝑡;𝑥)=𝑅_∞,𝑞(𝑡;𝑥)=𝑥,𝑅_𝑛,𝑞𝑡²;𝑥=𝑥²+𝑥(1−𝑥)[𝑛]−𝑥²(1−𝑥)1−𝑞11−𝑥+𝑥𝑞1−[𝑛]. (19)(20)

Lemma 10. With the definitions of𝑟_𝑛,𝑘(𝑞;𝑥) and𝑟_∞,𝑘(𝑞;𝑥), we have ^𝑛_𝑘=0𝑞^𝑘𝑟_𝑛,𝑘𝑞;𝑥=1−𝑥+𝑞^𝑛𝑥,^∞_𝑘=0𝑞^𝑘𝑟_∞,𝑘𝑞;𝑥=1−𝑥. (21)

Proof. Using (19) and (3), we get ^𝑛_𝑘=0𝑞^𝑘𝑟_𝑛,𝑘=𝑞𝑞;𝑥_𝑛−1^𝑛_𝑘=0𝑞^𝑘−1𝑞^𝑛𝑟−1^𝑛,𝑘𝑞;𝑥+^𝑛_𝑘=0𝑟_𝑛,𝑘=𝑞𝑞;𝑥_𝑛−1^𝑛_𝑘=0[𝑘][𝑛]𝑟_𝑛,𝑘=𝑞𝑞;𝑥+1_𝑛−1𝑅_𝑛,𝑞(𝑡;𝑥)+1=1−𝑥+𝑞^𝑛𝑥. (22) Similarly, using (19) and (5), we have ^∞_𝑘=0𝑞^𝑘𝑟_∞,𝑘=𝑞;𝑥_∞𝑘=0𝑞^𝑘−1𝑟_∞,𝑘𝑞;𝑥+^∞_𝑘=0𝑟_∞,𝑘𝑞;𝑥=−^∞_𝑘=01−𝑞^𝑘𝑟_∞,𝑘𝑞;𝑥+^∞_𝑘=0𝑟_∞,𝑘𝑞;𝑥=−𝑅_∞,𝑞(𝑡;𝑥)+1=1−𝑥. (23) The proof of Lemma 10 is complete.

For integers 𝑛,𝑘, and 𝑞∈(0,1), 𝑥∈[0,1], we have 𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞𝑘=𝑛𝑘𝑞𝑞;𝑥_{𝑘(𝑘−1)/2}(𝑥/(1−𝑥))^𝑘∏^𝑛−1_𝑠=01+𝑞^𝑠((−𝑞𝑥/1−𝑥))_{𝑘(𝑘−1)/2}(𝑥/(1−𝑥))^𝑘1−𝑞^𝑘[𝑘]!∏^∞_𝑠=01+𝑞^𝑠((=𝑛𝑘𝑞𝑥/1−𝑥))_{𝑘(𝑘−1)/2}(𝑥/(1−𝑥))^𝑘∏^𝑛−1_𝑠=01+𝑞^𝑠((×1𝑥/1−𝑥))1−∏^∞_𝑠=𝑛1+𝑞^𝑠((+𝑞𝑥/1−𝑥))_{𝑘(𝑘−1)/2}(𝑥/(1−𝑥))^𝑘∏^∞_𝑠=01+𝑞^𝑠((𝑛𝑘−1𝑥/1−𝑥))1−𝑞^𝑘[𝑘]!=𝑟_𝑛,𝑘1𝑞;𝑥1−∏^∞_𝑠=𝑛1+𝑞^𝑠(𝑥/(1−𝑥))−𝑟_∞,𝑘𝑞;𝑥1−^𝑛_{𝑠=𝑛−𝑘+1}1−𝑞^𝑠=𝑟_𝑛,𝑘𝑞;𝑥𝐽₁−𝑟_∞,𝑘𝑞;𝑥𝐽₂, (24) where 𝐽₁1∶=1−∏^∞_𝑠=𝑛1+𝑞^𝑠((,𝐽𝑥/1−𝑥))₂∶=1−^𝑛_{𝑠=𝑛−𝑘+1}1−𝑞^𝑠. (25) We will prove the following lemma.

Lemma 11. Let0<𝑞<1. Then for integers 𝑛,𝑘 and for0<𝑥<1/(1+𝑞^𝑛), ^𝑛_𝑘=0𝑞^𝑘𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞,𝑘𝑞;𝑥≤3𝑞^𝑛.1−𝑞 (26)

Proof. It is easy to prove by induction that 0≤𝐽₂∶=1−^𝑛_{𝑠=𝑛−𝑘+1}1−𝑞^𝑠≤^𝑛_{𝑠=𝑛−𝑘+1}𝑞^𝑠≤^∞_{𝑠=𝑛−𝑘}𝑞^𝑠=𝑞^𝑛−𝑘.1−𝑞 (27) Since 1−exp(−𝑥)≤𝑥 and ln(1+𝑥)≤𝑥 for all 𝑥∈[0,∞), we obtain 0≤𝐽₁=1−exp−^∞_𝑠=𝑛ln1+𝑞^𝑠𝑥≤1−𝑥_∞𝑠=𝑛ln1+𝑞^𝑠𝑥≤1−𝑥_∞𝑠=𝑛𝑞^𝑠𝑥=𝑞1−𝑥^𝑛𝑥).1−𝑞(1−𝑥 (28) Hence, 𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞𝑘≤𝑞𝑞;𝑥^𝑛𝑥)𝑟1−𝑞(1−𝑥^𝑛,𝑘+𝑞𝑞;𝑥_𝑛−𝑘𝑟1−𝑞^∞,𝑘𝑞;𝑥, (29) and therefore, by (21) and (19) we get ^𝑛_𝑘=0𝑞^𝑘𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞,𝑘≤𝑞𝑞;𝑥_𝑛𝑥1−𝑞(1−𝑥)^𝑛_𝑘=0𝑞^𝑘𝑟_𝑛,𝑘+𝑞𝑞;𝑥^𝑛1−𝑞^𝑛_𝑘=0𝑟_∞,𝑘≤𝑞𝑞;𝑥_𝑛𝑥1−𝑞(1−𝑥)1−𝑥+𝑞^𝑛𝑥+𝑞^𝑛.1−𝑞 (30) Since 0<𝑥<1/(1+𝑞^𝑛)<1, it follows that 0<𝑥/(1−𝑥)<1/𝑞^𝑛 and thence ^𝑛_𝑘=0𝑞^𝑘𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞,𝑘𝑞;𝑥≤3𝑞^𝑛.1−𝑞 (31) This completes the proof of Lemma 11.

Proof of Theorem 1. 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, 𝑅_𝑛,𝑞𝑓;0=𝑅_∞,𝑞(0),𝑅𝑓;0=𝑓_𝑛,𝑞𝑓;1=𝑅_∞,𝑞𝑓;1=𝑓(1). (32) 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 𝐼=𝑅_𝑛,𝑞𝑓;𝑥−𝑅_∞,𝑞=𝑓;𝑥_𝑛𝑘=0𝑓[𝑘][𝑛]−𝑓(1)𝑟_𝑛,𝑘−𝑞;𝑥_∞𝑘=0𝑓1−𝑞^𝑘−𝑓(1)𝑟_∞,𝑘≤𝑞;𝑥_𝑛𝑘=0𝑓[𝑘][𝑛]−𝑓(1)𝑟_𝑛,𝑘+𝑞;𝑥_∞𝑘=0𝑓1−𝑞^𝑘−𝑓(1)𝑟_∞,𝑘𝑞;𝑥. (33) Since [𝑘][𝑛]−1=1−𝑞^𝑘1−𝑞^𝑛≤𝑞−1^𝑘1−𝑞^𝑛−𝑘1−𝑞^𝑛≤𝑞^𝑘,(),𝜔≤()𝜔0≤𝑘≤𝑛𝑓;𝜆𝑡1+𝜆𝑓;𝑡,𝜆>0, (34) we get 𝐼≤^𝑛_𝑘=0𝜔𝑓;𝑞^𝑘𝑟_𝑛,𝑘𝑞;𝑥+^∞_𝑘=0𝜔𝑓;𝑞^𝑘𝑟_∞,𝑘≤𝑞;𝑥_𝑛𝑘=0𝜔𝑓,𝑞^𝑛𝑞1+^𝑘𝑞^𝑛𝑟_𝑛,𝑘+𝑞;𝑥_∞𝑘=0𝜔𝑓;𝑞^𝑛𝑞1+^𝑘𝑞^𝑛𝑟_∞,𝑘𝑞;𝑥≤2𝜔𝑓;𝑞^𝑛+𝜔𝑓,𝑞^𝑛𝑞^𝑛𝑛_𝑘=0𝑞^𝑘𝑟_𝑛,𝑘+𝜔𝑞;𝑥𝑓,𝑞^𝑛𝑞^𝑛∞_𝑘=0𝑞^𝑘𝑟_∞,𝑘𝑞;𝑥. (35) By Lemma 10 and 1−𝑥<𝑞^𝑛, 𝑥<1, we have 𝐼≤2𝜔𝑓;𝑞^𝑛+𝜔𝑓,𝑞^𝑛𝑞^𝑛1−𝑥+𝑞^𝑛𝑥+𝜔𝑓,𝑞^𝑛𝑞^𝑛(1−𝑥)≤5𝜔𝑓;𝑞^𝑛. (36)
Next, we assume that 0<𝑥<1/(1+𝑞^𝑛). Then 0≤𝑥/(1−𝑥)≤1/𝑞^𝑛. We have 𝐼=𝑅_𝑛,𝑞𝑓;𝑥−𝑅_∞,𝑞=𝑓;𝑥_𝑛𝑘=0𝑓[𝑘][𝑛]−𝑓1−𝑞^𝑘𝑟_𝑛,𝑘+𝑞;𝑥_𝑛𝑘=0𝑓1−𝑞^𝑘−𝑓(1)𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞,𝑘−𝑞;𝑥_{∞𝑘=𝑛+1}𝑓1−𝑞^𝑘−𝑓(1)𝑟_∞,𝑘≤𝑞;𝑥_𝑛𝑘=0𝑓[𝑘][𝑛]−𝑓1−𝑞^𝑘𝑟_𝑛,𝑘+𝑞;𝑥_𝑛𝑘=0𝑓1−𝑞^𝑘−𝑓(1)𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞,𝑘+𝑞;𝑥_{∞𝑘=𝑛+1}𝑓1−𝑞^𝑘−𝑓(1)𝑟_∞,𝑘𝑞;𝑥=∶𝛿₁+𝛿₂+𝛿₃. (37) First we estimate 𝛿₁ and 𝛿₃. Since [𝑘][𝑛]−1−𝑞^𝑘=1−𝑞^𝑘1−𝑞^𝑛−1−𝑞^𝑘=𝑞^𝑛1−𝑞^𝑘1−𝑞^𝑛≤𝑞^𝑛,(0≤𝑘≤𝑛)1−1−𝑞^𝑘=𝑞^𝑘≤𝑞^𝑛,(𝑘≥𝑛+1), (38) we get 𝛿₁≤𝜔𝑓,𝑞^𝑛𝑛_𝑘=0𝑟_𝑛,𝑘𝑞;𝑥=𝜔𝑓,𝑞^𝑛,𝛿₃≤𝜔𝑓,𝑞^𝑛∞_𝑘=𝑛+1𝑟_∞,𝑘𝑞;𝑥≤𝜔𝑓,𝑞^𝑛. (39)(40) Now we estimate 𝛿₂. Since 𝜔(𝑓,𝜆𝑡)≤(1+𝜆)𝜔(𝑓,𝑡), by Lemma 11 we get 𝛿₂≤^𝑛_𝑘=0𝜔𝑓,𝑞^𝑘𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞,𝑘≤𝑞;𝑥_𝑛𝑘=0𝜔𝑓,𝑞^𝑛𝑞1+^𝑘𝑞^𝑛𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞,𝑘≤𝑞;𝑥2𝜔𝑓;𝑞^𝑛𝑞^𝑛𝑛_𝑘=0𝑞^𝑘𝑟_𝑛,𝑘𝑞;𝑥−𝑟_∞,𝑘𝑞;𝑥≤6𝜔𝑓;𝑞^𝑛.1−𝑞 (41) From (39)–(41), we have for 0≤𝑥≤1/(1+𝑞^𝑛), 6𝐼≤2+𝜔1−𝑞𝑓;𝑞^𝑛. (42) Hence from (36) and (42), we conclude that, for 𝑞∈(0,1), 𝑅_𝑛,𝑞𝑓;𝑥−𝑅_∞,𝑞𝑓;𝑥≤𝐶_𝑞𝜔𝑓;𝑞^𝑛, (43) 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−𝑞²,1], 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)≤𝑐𝑡^𝛼, and 𝑅_𝑛,𝑞𝑓_𝛼−𝑅_∞,𝑞𝑓_𝛼≈𝑞^𝑛𝛼𝑟_𝑛,1𝑞;⋅≍𝑞^𝑛𝛼. (44) The proof of Theorem 1 is complete.

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

Theorem A (see [12]). Let the sequence {𝐿_𝑛} of positive linear operators on 𝐶[0,1] satisfy the following conditions.(A)The sequence {𝐿_𝑛(𝑒₂)} converges to a function 𝐿_∞(𝑒₂) in 𝐶[0,1], where 𝑒_𝑖(𝑥)=𝑥^𝑖, 𝑖=0,1,2.(B)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, 𝐿_𝑛𝑓,𝑥−𝐿_∞𝑓,𝑥≤𝑐𝜔₂𝑓;𝜆_𝑛(𝑥), (45) where 𝜆_𝑛(𝑥)=𝐿_𝑛(𝑒₂,𝑥)−𝐿_∞(𝑒₂,𝑥) and 𝑐 is a constant which depends only on ‖𝐿₁(𝑒₀)‖.

Proof of Theorem 2. From [2], we know that the Lupas 𝑞-analogues of the Bernstein operators satisfy Condition (B). It follows from [19] that, for 𝑞∈(0,1), {𝑅_𝑛,𝑞(𝑓;𝑥)} converges to 𝑅_∞,𝑞(𝑓;𝑥) uniformly in 𝑥∈[0,1] as 𝑛→∞; and 0≤𝜆_𝑛(𝑥)=𝑅_𝑛,𝑞𝑡²,𝑥−𝑅_∞,𝑞𝑡²,𝑥=𝑅_𝑛,𝑞𝑡²,𝑥−lim_𝑛→∞𝑅_𝑛,𝑞𝑡²=𝑥();𝑥1−𝑥[𝑛]−𝑥²(1−𝑥)1−𝑞11−𝑥+𝑥𝑞1−[𝑛]()+𝑥−𝑥1−𝑥1−𝑞²(1−𝑥)1−𝑞𝑞()11−𝑥+𝑥𝑞=𝑥1−𝑥[𝑛]−+𝑥1−𝑞₂(1−𝑥)1−𝑞11−𝑥+𝑥𝑞[𝑛]−=𝑥()1−𝑞1−𝑥𝑞1−𝑥+𝑥𝑞1−𝑞^𝑛1−𝑞^𝑛≤𝑞^𝑛. (46) 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).