On the Sets of Convergence for Sequences of the q-Bernstein Polynomials with q>1

and Applied Analysis 3 and leave the linear functions invariant: Bn,q at b;x ax b, n 1, 2, . . . , q > 0. 1.10 In addition, the q-Bernstein basic polynomials 1.2 satisfy the identity n ∑ k 0 pnk ( q;x ) 1 ∀n 1, 2, . . . , ∀q > 0. 1.11 Furthermore, the q-Bernstein polynomials admit a representation via the divided differences given by 3.3 , as well as demonstrate the saturation phenomenon see 2, 7, 13 . Despite the similarities such as those indicated above, the convergence properties of the q-Bernstein polynomials for q / 1 are essentially different from those of the classical ones. What is more, the cases 0 < q < 1 and q > 1 in terms of convergence are not similar to each other, as shown in 14, 15 . This absence of similarity is brought about by the fact that, for 0 < q < 1, Bn,q are positive linear operators on C 0, 1 , whereas for q > 1, no positivity occurs. In addition, the case q > 1 is aggravated by the rather irregular behavior of basic polynomials 1.2 , which, in this case, combine the fast increase in magnitude with the sign oscillations. For a detailed examination of this situation, see 16 , where, in particular, it has been shown that the norm ‖Bn,q‖ increases rather rapidly in both n and q. Namely, ∥Bn,q ∥ ∼ 2 e · q n n−1 /2 n as n −→ ∞, q −→ ∞. 1.12 This puts serious obstacles in the analysis of the convergence for q > 1. The challenge has inspired some papers by a number of authors dealing with the convergence of q-Bernstein polynomials in the case q > 1 see, e.g., 7, 17 . However, there are still many open problems related to the behavior of the q-Bernstein polynomials with q > 1 see the list of open problems in 2 . In this paper, it is shown that the time scale Jq { q−j }∞ j 0 ∪ {0} 1.13 is the “minimal” set of convergence for the q-Bernstein polynomials of continuous functions with q > 1, in the sense that every sequence {Bn,q f ;x } converges uniformly on Jq. Moreover, it is proved that Jq is the only set of convergence for some continuous functions. The paper is organized as follows. In Section 2, we present results concerning the convergence of the q-Bernstein polynomials on the time scale Jq. Section 3 is devoted to the q-Bernstein polynomials of the Weierstrass-type functions. Some of the results throughout the paper are also illustrated using numerical examples. 4 Abstract and Applied Analysis 2. The Convergence of the q-Bernstein Polynomials on Jq In this paper, q > 1 is considered fixed. It has been shown in 15 , that, if a function f is analytic in Dε {z : |z| < 1 ε}, then it is uniformly approximated by its q-Bernstein polynomials on any compact set in Dε, and, in particular, on 0, 1 . In this study, attention is focused on the q-Bernstein polynomials of “bad” functions, that is, functions which do not have an analytic continuation from 0, 1 to the unit disc. In general, such functions are not approximated by their q-Bernstein polynomials on 0, 1 . Moreover, their q-Bernstein polynomials may tend to infinity at some points of 0, 1 a simple example has been provided in 15 . Here, it is proved that the divergence of {Bn,q f ;x }may occur everywhere outside of Jq, which is a “minimal” set of convergence. However, in spite of this negative information, it will be shown that, for any f ∈ C 0, 1 , the sequence of its q-Bernstein polynomials converges uniformly on the time scale Jq. The next statement generalizing Lemma 1 of 15 can be regarded as a discrete analogue of the Popoviciu Theorem. Theorem 2.1. Let f ∈ C 0, 1 . Then ∣∣∣Bn,q ( f ; q−j ) − f ( q−j ∣∣∣ ≤ 2ωf ⎛ ⎝ √√ √q −j1 − q−j n q ⎞ ⎠, j ∈ Z , 2.1 where ωf is the modulus of continuity of f on 0, 1 . Corollary 2.2. If j ∈ Z , then ∣∣∣Bn,q ( f ; q−j ) − f ( q−j ∣∣ ≤ 2ωf ⎛ ⎜ ⎝ 1 2 √ n q ⎞ ⎟ ⎠, 2.2 that is, Bn,q f ;x converges uniformly to f x on the time scale Jq. Proof. The proof is rather straightforward. First, notice that pnk q; q−j ≥ 0 for all n, k, j, while ∑n k 0 pnk q; q −j 1 by virtue of 1.11 . Then ∣∣∣Bn,q ( f ; q−j ) − f ( q−j ∣∣ ≤ n ∑ k 0 ∣∣∣∣ f ( k q n q ) − f ( q−j ) ∣∣∣∣ pnk ( q; q−j ) ≤ n ∑ k 0 ωf (∣∣∣∣ k q n q − q−j ∣∣∣∣ ) pnk ( q; q−j ) ≤ ωf δ n ∑ k 0 ⎧ ⎨ ⎩ 1 1 δ2 ( k q n q − q−j )2 ⎫ ⎬ ⎭ pnk ( q; q−j ) 2.3 Abstract and Applied Analysis 5 for any δ > 0. Plain calculations see, e.g., 13 , formula 2.7 show thatand Applied Analysis 5 for any δ > 0. Plain calculations see, e.g., 13 , formula 2.7 show that Bn,q ( t − x ;x ) x 1 − x n q , 2.4 which implies that ∣ ∣ ∣Bn,q ( f ; q−j ) − f ( q−j )∣ ∣ ≤ ωf δ · { 1 1 δ2 · q −j1 − q−j n q } . 2.5 Then, one can immediately derive the result by choosing δ √ q−j 1 − q−j / n q. Remark 2.3. In 7 , Wu has shown that if f ∈ C1 0, 1 , then for any j ∈ Z , one has: ∣∣∣Bn,q ( f ; q−j ) − f ( q−j ∣∣ ≤ Cj ( q−n ) , n −→ ∞, where Cj −→ ∞ as j → ∞. 2.6 The condition f ∈ C1 0, 1 cannot be left out completely, as the following example shows. Example 2.4. Consider a function f ∈ C 0, 1 satisfying f x ⎧ ⎪⎨ ⎪⎩ 0 if x ∈ 0, q−2 ∪ q−1, 1, ( q−1 − xα if x ∈ [( q−2 q−1 ) 2 , q−1 ] , 2.7 where 0 < α < 1. Then, for n large enough, we have Bn,q ( f ; q−1 ) − f ( q−1 ) f ( n − 1 q n q ) pn,n−1 ( q; q−1 ) ( q − 1 q )α · ( q − 11−α qn ≥ C q−nα, 2.8 where C is a positive constant independent from n. As it has been already mentioned, the behavior of the q-Bernstein polynomials in the case q > 1 outside of the time scale Jq may be rather unpredictable. The next theorem shows that the sequence {Bn,q f ;x }may be divergent for all x ∈ R \ Jq. Theorem 2.5. Let f x x, 0 < α ≤ 1/2. If q ≥ 2, then Bn,q ( f ;x ) → ∞ as n → ∞ ∀x ∈ R \ Jq. 2.9 6 Abstract and Applied Analysis Proof. The q-Bernstein polynomial of f is Bn,q ( f ;x ) n ∑ k 0 ( k q n q )α pnk ( q;x ) 1 n q n ∑ k 1 k αq pnk ( q;x ) . 2.10 Since for k 1, 2, . . . , n − 1 one has pnk ( q;x ) ( q − 1 · · · qn−k 1 − 1 ( qk − 1 · · · q − 1 x kx; q ) n−k q 2n−k 1 k/2 ( q−n; q ) k ( qk − 1 · · · q − 1 −1 n−kq n−k n−k−1 /2xn ( 1 x ; 1 q ) n−k −1 q n−1 /2xn · −1 qk ( q−n; q ) k ( qk − 1 · · · q − 1 · ( 1 x ; 1 q ) n−k , 2.11


Introduction
Let f : 0, 1 → C, q > 0, and n ∈ N. Then, the q-Bernstein polynomial of f is defined by Here, for any nonnegative integer k, k q ! 1 q 2 q · · · k q k 1, 2, . . . , 0 q ! : 1 1.5 are the q-factorials with k q being the q-integer given by k q 1 q · · · q k−1 k 1, 2, . . . , 0 q : 0. 1.6 We use the notation from 1 , Ch. 10 . The polynomials p n0 q; x , p n1 q; x , . . . , p nn q; x , called the q-Bernstein basic polynomials, form the q-Bernstein basis in the linear space of polynomials of degree at most n.
Although, for q 1, the q-Bernstein polynomial B n,q f; x turns into the classical Bernstein polynomial B n f; x : conventionally, the name "q-Bernstein polynomials" is reserved for the case q / 1. Based on the q-Bernstein polynomials, the q-Bernstein operator on C 0, 1 is given by B n,q : f −→ B n,q f; · . 1.8 A detailed review of the results on the q-Bernstein polynomials along with an extensive bibliography has been provided in 2 . In this field, new results concerning the properties of the q-Bernstein polynomials and/or their various generalizations are still coming out see, e.g, papers 3-8 , all of which have appeared after 2 .
The popularity of the q-Bernstein polynomials is attributed to the fact that they are closely related to the q-binomial and the q-deformed Poisson probability distributions cf. 9 . The q-binomial distribution plays an important role in the q-boson theory, providing a q-deformation for the quantum harmonic formalism. More specifically, it has been used to construct the binomial state for the q-boson. Meanwhile, the q-deformed Poisson distribution, which is the limit form of q-binomial one, defines the energy distribution in a q-analogue of the coherent state 10 . Another motivation for this study is that various estimates related to the natural sequences of functions and operators in functional spaces, convergence theorems, and estimates for the rates of convergence are of decisive nature in the modern functional analysis and its applications see, e.g., 4, 11, 12 . The q-Bernstein polynomials retain some of the properties of the classical Bernstein polynomials. For example, they possess the end-point interpolation property: Furthermore, the q-Bernstein polynomials admit a representation via the divided differences given by 3.3 , as well as demonstrate the saturation phenomenon see 2, 7, 13 .
Despite the similarities such as those indicated above, the convergence properties of the q-Bernstein polynomials for q / 1 are essentially different from those of the classical ones. What is more, the cases 0 < q < 1 and q > 1 in terms of convergence are not similar to each other, as shown in 14, 15 . This absence of similarity is brought about by the fact that, for 0 < q < 1, B n,q are positive linear operators on C 0, 1 , whereas for q > 1, no positivity occurs. In addition, the case q > 1 is aggravated by the rather irregular behavior of basic polynomials 1.2 , which, in this case, combine the fast increase in magnitude with the sign oscillations. For a detailed examination of this situation, see 16 , where, in particular, it has been shown that the norm B n,q increases rather rapidly in both n and q. Namely, B n,q ∼ 2 e · q n n−1 /2 n as n −→ ∞, q −→ ∞.

1.12
This puts serious obstacles in the analysis of the convergence for q > 1. The challenge has inspired some papers by a number of authors dealing with the convergence of q-Bernstein polynomials in the case q > 1 see, e.g., 7, 17 . However, there are still many open problems related to the behavior of the q-Bernstein polynomials with q > 1 see the list of open problems in 2 .
In this paper, it is shown that the time scale is the "minimal" set of convergence for the q-Bernstein polynomials of continuous functions with q > 1, in the sense that every sequence {B n,q f; x } converges uniformly on J q . Moreover, it is proved that J q is the only set of convergence for some continuous functions. The paper is organized as follows. In Section 2, we present results concerning the convergence of the q-Bernstein polynomials on the time scale J q . Section 3 is devoted to the q-Bernstein polynomials of the Weierstrass-type functions. Some of the results throughout the paper are also illustrated using numerical examples.

The Convergence of the q-Bernstein Polynomials on J q
In this paper, q > 1 is considered fixed. It has been shown in 15 , that, if a function f is analytic in D ε {z : |z| < 1 ε}, then it is uniformly approximated by its q-Bernstein polynomials on any compact set in D ε , and, in particular, on 0, 1 .
In this study, attention is focused on the q-Bernstein polynomials of "bad" functions, that is, functions which do not have an analytic continuation from 0, 1 to the unit disc. In general, such functions are not approximated by their q-Bernstein polynomials on 0, 1 . Moreover, their q-Bernstein polynomials may tend to infinity at some points of 0, 1 a simple example has been provided in 15 . Here, it is proved that the divergence of {B n,q f; x } may occur everywhere outside of J q , which is a "minimal" set of convergence.
However, in spite of this negative information, it will be shown that, for any f ∈ C 0, 1 , the sequence of its q-Bernstein polynomials converges uniformly on the time scale J q .
The next statement generalizing Lemma 1 of 15 can be regarded as a discrete analogue of the Popoviciu Theorem.
where ω f is the modulus of continuity of f on 0, 1 .
that is, B n,q f; x converges uniformly to f x on the time scale J q .
Proof. The proof is rather straightforward. First, notice that p nk q; q −j ≥ 0 for all n, k, j, while n k 0 p nk q; q −j 1 by virtue of 1.11 . Then Abstract and Applied Analysis 5 for any δ > 0. Plain calculations see, e.g., 13 , formula 2.7 show that Then, one can immediately derive the result by choosing δ Remark 2.3. In 7 , Wu has shown that if f ∈ C 1 0, 1 , then for any j ∈ Z , one has: The condition f ∈ C 1 0, 1 cannot be left out completely, as the following example shows.
where 0 < α < 1. Then, for n large enough, we have where C is a positive constant independent from n.
As it has been already mentioned, the behavior of the q-Bernstein polynomials in the case q > 1 outside of the time scale J q may be rather unpredictable. The next theorem shows that the sequence {B n,q f; x } may be divergent for all x ∈ R \ J q .

2.14
As such, the theorem will be proved if it is shown that As lim n → ∞ q −n n−1 /2 −1 n n α q 0, it suffices to prove that Abstract and Applied Analysis 7 where c kn :

2.17
The fact that q −n ; q k ≤ 1 and the inequality and the series ∞ k 0 d k is convergent, the Lebesgues dominated convergence theorem implies How about the sum of the series in 2.21 ? Consider the following two cases.
Abstract and Applied Analysis for k ≥ 2 it follows that

2.24
Notice that 2.24 holds for any α ∈ 0, 1 . In addition, if k 1, then The function in the r.h.s. is monotone decreasing in q, so To prove the theorem, it suffices to show that a 1 − a 2 a 3 − a 4 > 0 when q ≥ 2. Denoting a i q q − 1 α / q − 1 g i q , i 1, 2, 3, 4, we write the following: We are left to show that K q is strictly positive for the specified values of q and α. First of all, notice that g 1 q 1, while g 2 q , g 3 q , and g 4 q are strictly decreasing in q on 0, ∞ . Hence, for q ∈ 2, 5/2 , The function L α is strictly decreasing on 1/3, 1/2 . Indeed, Similarly, for q ∈ 5/2, 3 ,
Remark 2.6. It can be seen from the proof that, the statement of the theorem is true for any α ∈ 0, 1 and q ≥ q 0 α .
An illustrative example is supplied below.
The graphs of y f x and y B n,q f; x for q 2 and n 4, 5 are exhibited in Figure 1. Similarly, Figure 2 represents the graphs of y f x and y B n,q f; x for q 2 and n 6, 7 over the subintervals 0, 0.5 and 0.5, 1 , respectively. In addition, Table 1 presents the values of the error function E n, q, x : B n,q f; x − f x with q 2 at some points x ∈ 0, 1 . The points are taken both in J q and in 0, 1 \ J q . It can be observed from Table 1 that, while at the points x ∈ J q , the values of the error function are close to 0, at the points x / ∈ J q , the values of the error function may be very large in magnitude.
Remark 2.8. Table 1 also shows that while the error function changes its sign for different values of x, for x q −j ∈ J q , its values are negative, that is, B n,q t 1/3 ; q −j < f q −j for q −j ∈ J q . This is a particular case of the following statement.
Proof. It can be readily seen from 1.10 and 1.11 that The function is concave on 0, 1 and, hence, according to the previous results, B n,q f; q −j → f q −j as n → ∞ from below for all j ∈ Z . To examine the behavior of polynomials B n,q f; x for x / ∈ J q , consider the auxiliary function: Since n − k q / n q ≤ q −k for k 0, 1, . . . , n, and n − 1 q / n q ≥ q −2 whenever q n ≥ q 1, it follows that, for sufficiently large n, B n,q g; x g n − 1 q n q p n,n−1 q; x g 1 p nn x .

2.42
Consequently, for x / ∈ J q , one obtains

2.45
Since, by 1.10 , B n,q f; x q 2 x B n,q g; x , it follows that:

2.46
For x −q −1 , the limit does not exist. Additionally, it is not difficult to see that B n,q f; x → f x as n → ∞ uniformly on any compact set inside −1/q 2 , 1/q 2 , while on any interval outside of −1/q 2 , 1/q 2 , the function f x is not approximated by its q-Bernstein polynomials. This agrees with the result from 17 , Theorem 2.3. The graphs of f x and B n,q f; x for q 2, n 5 and 8 on 0, 1 are given in Figure 3. The values of the error function at some points x ∈ J q and at some exemplary points x / ∈ J q are given in Table 2.
Remark 2.11. Following Charalambides 9 , consider a sequence of random variables {X j n } ∞ n 1 possessing the distributions P j n given by P X j n n − k q n q p n,n−k q; q −j , k 0, 1, . . . , n.

2.47
Let I q −j denote a random variable with the δ-distribution concentrated at q −j . Theorem 2.1 implies that X j n → I q −j in distribution.
Generally speaking, Theorem 2.1 shows that the q-Bernstein polynomials with q > 1 possess an "interpolation-type" property on J q . Information on interpolation of functions with nodes on a geometric progression can be found in, for example, 18 by Schoenberg.

On the q-Bernstein Polynomials of the Weierstrass-Type Functions
In this section, the q-Bernstein polynomials of the functions with "bad" smoothness are considered. Let ϕ x ∈ C −1, 1 satisfy the condition: The letter ϕ will also denote a 2-periodic continuation of ϕ x on −∞, ∞ .
Notice that f x is continuous if and only if ϕ −1 ϕ 1 . For ϕ x cos πx and a special choice of a and b see, e.g., 19, Section 4 , the classical Weierstrass continuous nowhere differentiable function is obtained. In 19 , one can also find an exhaustive bibliography on this function and similar ones. For ϕ x 1−|x|, a function analogous to the Van der Waerden continuous nowhere differentiable function appears.
The aim of this section is to prove the following statement.

Theorem 3.2.
If f x is a Weierstrass-type function, then the sequence B n,q f; x of its q-Bernstein polynomials is not uniformly bounded on any interval 0, c .
For n > b, such a choice is possible because, in this case, inequality 3.9 implies that 0 < ln n q ln b − 1 < j n ≤ ln n q ln b .

3.10
Since the length of the interval ln n q / ln b − 1, ln n q / ln b is 1, there is a positive integer, say, j n , such that j n ∈ ln n q / ln b−1, ln n q / ln b . The obvious inequality n q > q n−1 implies the following: with A ln q/ ln b 1 being a positive constant. Then, for n > b, it follows that B n,q f; 0 ≥ n q a j n min t∈ 1/b,1 ϕ 0 − ϕ t : τ n q a j n , 3.12 where τ > 0 due to 3.1 . Consequently, B n,q f; 0 ≥ τ n q b j n ab j n ≥ τ ab j n ≥ τ ab n ln q/ ln b −A , 3.13 which leads to B n,q f; 0 ≥ Cρ n , 3.14 where C τ ab −A is a positive constant and ρ ab ln q/ ln b > 1. Now, assume that {B n,q f; x } is uniformly bounded on 0, c , that is, |B n,q f; x | ≤ M for all x ∈ 0, c . By Markov's Inequality cf., e.g., 22 , Chapter 4, Section 1, pp. 97-98 it follows that B n,q f; 0 ≤ 2M c n 2 ∀n 1, 2, . . . , 3.15 This proves the theorem because the latter estimate contradicts 3.14 .
To present an illustrative example, let us denote the Nth partial sum of the series in 3.2 by h N , that is: 3.16 Clearly, the function h N is an approximation of 3.2 satisfying the error estimate