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

Authors: Sofiya Ostrovska; Ahmet Yaşar Özban
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The aim of this paper is to present new results related to the convergence of the sequence of theq-Bernstein polynomials {Bn,q(f;x)}  in the case q>1, where f is a continuous function on [0,1]. It is shown that the polynomials converge to f uniformly on the time scale 𝕁q={q-j}j=0∞∪{0}, and that this result is sharp in the sense that the sequence {Bn,q(f;x)}n=1∞  may be divergent for all x∈R∖𝕁q. Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.


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## Body

## 1. Introduction

Letf:[0,1]→ℂ, q>0, and n∈ℕ. Then, the q-Bernstein polynomial of f is defined by
(1.1)Bn,q(f;x)=∑k=0nf([k]q[n]q)pnk(q;x),
where
(1.2)pnk(q;x)=[nk]qxk(x;q)n-k,k=0,1,…n,
with [nk]q being the q-binomial coefficients given by
(1.3)[nk]q=[n]q![k]q![n-k]q!,
and (x;q)m being the q-Pochhammer symbol:
(1.4)(x;q)0=1,(x;q)m=∏s=0m-1(1-xqs),(x;q)∞=∏s=0∞(1-xqs).
Here, for any nonnegative integer k,
(1.5)[k]q!=[1]q[2]q⋯[k]q(k=1,2,…),[0]q!:=1
are the q-factorials with [k]q being the q-integer given by
(1.6)[k]q=1+q+⋯+qk-1(k=1,2,…),[0]q:=0.
We use the notation from [[1], Ch. 10].The polynomialspn0(q;x),pn1(q;x),…,pnn(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, forq=1, the q-Bernstein polynomial Bn,q(f;x) turns into the classical Bernstein polynomial Bn(f;x):
(1.7)Bn(f;x)=∑k=0nf(kn)(nk)xk(1-x)n-k,
conventionally, the name “q-Bernstein polynomials” is reserved for the case q≠1.Based on theq-Bernstein polynomials, the q-Bernstein operator on C[0,1] is given by
(1.8)Bn,q:f↦Bn,q(f;·).
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 theq-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]).Theq-Bernstein polynomials retain some of the properties of the classical Bernstein polynomials. For example, they possess the end-point interpolation property:
(1.9)Bn,q(f;0)=f(0),Bn,q(f;1)=f(1),n=1,2,…,q>0,
and leave the linear functions invariant:
(1.10)Bn,q(at+b;x)=ax+b,n=1,2,…,q>0.
In addition, the q-Bernstein basic polynomials (1.2) satisfy the identity
(1.11)∑k=0npnk(q;x)=1∀n=1,2,…,∀q>0.
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 theq-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,
(1.12)∥Bn,q∥~2e·qn(n-1)/2nasn→∞,q→+∞.
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(1.13)𝕁q={q-j}j=0∞∪{0}
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 Section2, 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.


## 2. The Convergence of theq-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 theq-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 anyf∈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.
Letf∈C[0,1]. Then
(2.1)|Bn,q(f;q-j)-f(q-j)|≤2ωf(q-j(1-q-j)[n]q),j∈ℤ+,
where ωf is the modulus of continuity of f on [0,1].Corollary 2.2.
Ifj∈ℤ+, then
(2.2)|Bn,q(f;q-j)-f(q-j)|≤2ωf(12[n]q),
that is, Bn,q(f;x) converges uniformly to f(x) on the time scale 𝕁q.Proof.
The proof is rather straightforward. First, notice thatpnk(q;q-j)≥0 for all n,k,j, while ∑k=0n‍pnk(q;q-j)=1 by virtue of (1.11). Then
(2.3)|Bn,q(f;q-j)-f(q-j)|≤∑k=0n‍|f([k]q[n]q)-f(q-j)|pnk(q;q-j)≤∑k=0n‍ωf(|[k]q[n]q-q-j|)pnk(q;q-j)≤ωf(δ)∑k=0n‍{1+1δ2([k]q[n]q-q-j)2}pnk(q;q-j)
for any δ>0. Plain calculations (see, e.g., [13], formula (2.7)) show that
(2.4)Bn,q((t-x)2;x)=x(1-x)[n]q,
which implies that
(2.5)|Bn,q(f;q-j)-f(q-j)|≤ωf(δ)·{1+1δ2·q-j(1-q-j)[n]q}.
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:
(2.6)|Bn,q(f;q-j)-f(q-j)|≤Cj(q-n),n→∞,whereCj→∞asj→∞.
The condition f∈C1[0,1] cannot be left out completely, as the following example shows.Example 2.4.
Consider a functionf∈C[0,1] satisfying
(2.7)f(x)={0ifx∈[0,q-2]∪(q-1,1],(q-1-x)αifx∈[(q-2+q-1)2,q-1],
where 0<α<1. Then, for n large enough, we have
(2.8)Bn,q(f;q-1)-f(q-1)=f([n-1]q[n]q)pn,n-1(q;q-1)=(q-1q)α·(qn-1)1-αqn≥Cq-nα,
where C is a positive constant independent from n.As it has been already mentioned, the behavior of theq-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.
Letf(x)=xα,  0<α≤1/2. If q≥2, then
(2.9)Bn,q(f;x)→∞asn→∞∀x∈ℝ∖𝕁q.Proof.
Theq-Bernstein polynomial of f is
(2.10)Bn,q(f;x)=∑k=0n([k]q[n]q)αpnk(q;x)=1[n]qα∑k=1n‍[k]qαpnk(q;x).
Since for k=1,2,…,n-1 one has
(2.11)pnk(q;x)=(qn-1)⋯(qn-k+1-1)(qk-1)⋯(q-1)xk(x;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(1x;1q)n-k=(-1)nqn(n-1)/2xn·(-1)kqk(q-n;q)k(qk-1)⋯(q-1)·(1x;1q)n-k,
it follows that
(2.12)Bn,q(f;x)=(-1)nqn(n-1)/2xn[n]qα·T(n,q,x),
where
(2.13)T(n;q;x):=(-1)n[n]qαqn(n-1)/2+∑k=0n-1(-1)k[k]qαqk(q-n;q)k(qk-1)⋯(q-1)·(1x;1q)n-k.
Obviously,
(2.14)limn→∞qn(n-1)/2xn[n]qα=∞foranyx≠0.
As such, the theorem will be proved if it is shown that
(2.15)limn→∞T(n,q,x)≠0forx∉𝕁q.
As limn→∞(q-n(n-1)/2(-1)n[n]qα)=0, it suffices to prove that
(2.16)limn→∞∑k=0∞ckn≠0whenx∉𝕁q,
where
(2.17)ckn:={(-1)k[k]qαqk(q-n;q)k(qk-1)⋯(q-1)·(1x;1q)n-kifk≤n-1,0ifk≥n.
The fact that (q-n;q)k≤1 and the inequality
(2.18)|(1x;1q)n-k|≤(-1|x|;1q)n-k≤(-1|x|;1q)∞
lead to
(2.19)|ckn|≤qk[k]qα(qk-1)⋯(q-1)(-1|x|;1q)∞=:dk.
Now, since
(2.20)limn→∞ckn=(-1)kqk[k]qα(qk-1)⋯(q-1)(1x;1q)∞,
and the series ∑k=0∞‍dk is convergent, the Lebesgues dominated convergence theorem implies
(2.21)limn→∞∑k=0∞‍ckn=∑k=0∞limn→∞ckn=1(q-1)α(1x;1q)∞·∑k=0∞(-1)kak,
where ak=qk(qk-1)α/(qk-1)⋯(q-1),k=1,2,…. Moreover,
(2.22)1(q-1)α(1x;1q)∞≠0wheneverx∉𝕁q.
How about the sum of the series in (2.21)? Consider the following two cases.
Case
1. 0<α<1/3.
Let us show thatak+1<ak,  k=1,2,… for q≥2. Since
(2.23)ak+1ak=q(qk+1-1)1-α(qk-1)α,
for k≥2 it follows that
(2.24)ak+1ak≤q(q3-1)1-α(q2-1)α≤q(q-1)(q2+q+1)1-α(q+1)α≤q(q-1)(q+1)<1.
Notice that (2.24) holds for any α∈(0,1). In addition, if k=1, then
(2.25)a2a1=q(q2-1)1-α(q-1)α=q(q-1)(q+1)1-α.
The function in the r.h.s. is monotone decreasing in q, so
(2.26)a2a1≤21·31-α≤293<1.
Thus, {ak}k=1∞ is a strictly decreasing sequence. Since all (a2k-1-a2k) are strictly positive, it follows that
(2.27)∑k=0∞(-1)kak=-[(a1-a2)+(a3-a4)+⋯+(a2k-1-a2k)+⋯]<0.
Case
2. 1/3≤α≤1/2.
Estimate (2.24) implies that ∑k=5∞‍(-1)kak<0. To prove the theorem, it suffices to show that a1-a2+a3-a4>0 when q≥2. Denoting ai=(q(q-1)α/(q-1))gi(q), i=1,2,3,4, we write the following:
(2.28)a1-a2+a3-a4=q(q-1)αq-1[g1(q)-g2(q)+g3(q)-g4(q)]=:q(q-1)αq-1K(q).
We are left to show that K(q) is strictly positive for the specified values of q and α. First of all, notice that g1(q)=1, while g2(q),g3(q), and g4(q) are strictly decreasing in q on (0,+∞). Hence, for q∈[2,5/2],
(2.29)K(q)≥1-g1(2)+g2(52)-g3(2)=1-23·3α+2002457(394)α-8315·15α=:L(α).
The function L(α) is strictly decreasing on [1/3,1/2]. Indeed,
(2.30)L′(α)=-23·3αln3+2002457(394)αln(394)-8315·15αln15
and, for α∈[1/3,1/2],
(2.31)L′(α)≤-23·31/3ln3+2002457(394)1/2ln(394)-8315·151/3ln15≤-0.4332<0,
whence L(α)≥L(1/2)≥1.096×10-3>0 for α∈[1/3,1/2].
Similarly, forq∈[5/2,3],
(2.32)K(q)≥1-g1(52)+g2(3)-g3(52)=1-1021·(72)α+9208·13α-80001496313·(2038)α=:M(α).
Applying the same reasoning as done for L(α), it can be shown that M(α) is strictly decreasing on [1/3,1/2]. Since M(1/2)≥0.238>0, it follows that M(α)>0 for all α∈[1/3,1/2].
Finally, forq∈[3,+∞), we obtain
(2.33)K(q)≥1-g1(3)-g3(3)=1-38·4α-2716640·40α=:N(α).
Obviously, N(α) is a strictly decreasing function for all α∈ℝ, whence, for α∈[1/3,1/2],
(2.34)N(α)≥N(12)≥0.239>0,
which completes the proof.Remark 2.6.
It can be seen from the proof that, the statement of the theorem is true for anyα∈(0,1) and q≥q0(α).An illustrative example is supplied below.Example 2.7.
Letf(x)=x3.  The graphs of y=f(x) and y=Bn,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=Bn,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 functionE(n,q,x):=Bn,q(f;x)-f(x) with q=2 at some points x∈[0,1]. The points are taken both in 𝕁q and in [0,1]∖𝕁q. It can be observed from Table 1 that, while at the points x∈𝕁q, the values of the error function are close to 0, at the points x∉𝕁q, the values of the error function may be very large in magnitude.Table 1
The values ofE(n,q,x)=Bn,q(f;x)-f(x) at some points x∈[0,1].
x
E
(
3,2
,
x
)
E
(
4,2
,
x
)
E
(
10,2
,
x
)
E
(
15,2
,
x
)
E
(
20,2
,
x
)
E
(
25,2
,
x
)
E
(
30,2
,
x
)
(
q
+
1
)
/
2
q
-
8.76
×
1
0
-
2
0.336
7
.
07
×
1
0
9
-
6.08
×
1
0
26
1.76
×
1
0
51
-
1.71
×
1
0
83
5.55
×
1
0
122
1
/
q
-
9.00
×
1
0
-
3
-
4.02
×
1
0
-
3
-
5.70
×
1
0
-
5
-
1.78
×
1
0
-
6
-
5.56
×
1
0
-
8
-
1.74
×
1
0
-
9
-
5.43
×
1
0
-
11
(
q
+
1
)
/
2
q
2
9.88
×
1
0
-
3
-
4.63
×
1
0
-
2
-
1.15
×
1
0
7
3.09
×
1
0
22
-
2.79
×
1
0
45
8.47
×
1
0
75
-
8.62
×
1
0
113
1
/
q
2
-
2.39
×
1
0
-
2
-
9.99
×
1
0
-
3
-
1.36
×
1
0
-
4
-
4.23
×
1
0
-
6
-
1.32
×
1
0
-
7
-
4.13
×
1
0
-
9
-
1.29
×
1
0
-
10
(
q
+
1
)
/
2
q
3
-
6.66
×
1
0
-
2
1.39
×
1
0
-
3
4.90
×
1
0
4
-
4.09
×
1
0
18
1.16
×
1
0
40
-
1.09
×
1
0
69
3.48
×
1
0
105
1
/
q
3
-
0.126
-
2.05
×
1
0
-
2
-
2.52
×
1
0
-
4
-
7.84
×
1
0
-
6
-
2.45
×
1
0
-
7
-
7.66
×
1
0
-
9
-
2.39
×
1
0
-
10
(
q
+
1
)
/
2
q
4
-
0.159
-
5.06
×
1
0
-
2
−465.
1.21
×
1
0
15
-
1.06
×
1
0
35
3.15
×
1
0
62
-
3.13
×
1
0
97
1
/
q
4
-
0.190
-
9.47
×
1
0
-
2
-
4.29
×
1
0
-
4
-
1.33
×
1
0
-
5
-
4.17
×
1
0
-
7
-
1.30
×
1
0
-
8
-
4.07
×
1
0
-
10
(
q
+
1
)
/
2
q
5
-
0.201
-
0.12
9.37
-
7.49
×
1
0
11
2.06
×
1
0
30
-
1.91
×
1
0
56
5.93
×
1
0
89
1
/
q
5
-
0.206
-
0.145
-
7.07
×
1
0
-
4
-
2.19
×
1
0
-
5
-
6.84
×
1
0
-
7
-
2.14
×
1
0
-
8
-
6.68
×
1
0
-
10
(
q
+
1
)
/
2
q
6
-
0.204
-
0.155
-
0.393
9.54
×
1
0
8
-
8.21
×
1
0
25
2.37
×
1
0
50
-
2.30
×
1
0
82
1
/
q
6
-
0.194
-
0.16
-
1.15
×
1
0
-
3
-
3.53
×
1
0
-
5
-
1.10
×
1
0
-
6
-
3.45
×
1
0
-
8
-
1.08
×
1
0
-
9
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
1
/
q
19
-
1.24
×
1
0
-
2
-
1.24
×
1
0
-
2
-
1.22
×
1
0
-
2
-
1.05
×
1
0
-
2
-
5.43
×
1
0
-
4
-
1.43
×
1
0
-
5
-
4.45
×
1
0
-
7
(
q
+
1
)
/
2
q
20
-
1.13
×
1
0
-
2
-
1.13
×
1
0
-
2
-
1.11
×
1
0
-
2
-
9.82
×
1
0
-
3
-
1.21
×
1
0
-
3
1.27
-
2.23
×
1
0
10
1
/
q
20
-
9.84
×
1
0
-
3
-
9.84
×
1
0
-
3
-
9.75
×
1
0
-
3
-
8.87
×
1
0
-
3
-
2.24
×
1
0
-
3
-
2.28
×
1
0
-
5
-
7.06
×
1
0
-
7
(
q
+
1
)
/
2
q
21
-
8.94
×
1
0
-
3
-
8.94
×
1
0
-
3
-
8.87
×
1
0
-
3
-
8.22
×
1
0
-
3
-
2.86
×
1
0
-
3
-
0.102
2.91
×
1
0
7
1
/
q
21
-
7.81
×
1
0
-
3
-
7.81
×
1
0
-
3
-
7.76
×
1
0
-
3
-
7.33
×
1
0
-
3
-
3.49
×
1
0
-
3
-
5.51
×
1
0
-
2
-
1.12
×
1
0
-
6
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
1
/
q
49
-
1.21
×
1
0
-
5
-
1.21
×
1
0
-
5
-
1.21
×
1
0
-
5
-
1.21
×
1
0
-
5
-
1.21
×
1
0
-
5
-
1.21
×
1
0
-
5
-
1.21
×
1
0
-
5
(
q
+
1
)
/
2
q
50
-
1.10
×
1
0
-
5
-
1.10
×
1
0
-
5
-
1.10
×
1
0
-
5
-
1.10
×
1
0
-
5
-
1.10
×
1
0
-
5
-
1.10
×
1
0
-
5
-
1.10
×
1
0
-
5
1
/
q
50
-
9.61
×
1
0
-
6
-
9.61
×
1
0
-
6
-
9.61
×
1
0
-
6
-
9.61
×
1
0
-
6
-
9.61
×
1
0
-
6
-
9.61
×
1
0
-
6
-
9.61
×
1
0
-
6Figure 1
Graphs ofy=f(x) and y=Bn,2(f;x),  n=4,5.Figure 2
Graphs ofy=f(x) and y=Bn,2(f;x),  n=6,7.Remark 2.8.
Table1 also shows that while the error function changes its sign for different values of x, for x=q-j∈𝕁q, its values are negative, that is, Bn,q(t1/3;q-j)<f(q-j) for q-j∈𝕁q. This is a particular case of the following statement.Theorem 2.9.
Letq>1. If f(x) is convex (concave) on [0,1], then
(2.35)Bn,q(f;q-j)≥f(q-j)(correspondinglyBn,q(f;q-j)≤f(q-j)),
for all q-j∈𝕁q.Proof.
It can be readily seen from (1.10) and (1.11) that
(2.36)∑k=0n‍pnk(q;q-j)=1,∑k=0n‍[k]q[n]qpnk(q;q-j)=q-j,
while pnk(q;q-j)≥0. By virtue of Jensen's inequality, if f is convex on [0,1], then whenever n∈ℕ and x0,x1,…,xn∈[a,b], there holds the following:
(2.37)∑k=0n‍λkf(xk)≥f(∑k=0n‍λkxk).
for all λ0,λ1,…,λn≥0 satisfying ∑k=0n‍λk=1. Setting
(2.38)xk=[k]q[n]q,λk=pnk(q;q-j),k=0,1,…,n,
and observing that
(2.39)∑k=0n‍λkf(xk)=Bn,q(f;q-j),
the required result is derived.Example 2.10.
Let(2.40)f(x)={q2xif0≤x≤q-2,q2x-(q2x-1)2q2-1ifq-2<x≤1.
The function is concave on [0,1] and, hence, according to the previous results, Bn,q(f;q-j)→f(q-j) as n→∞ from below for all j∈ℤ+. To examine the behavior of polynomials Bn,q(f;x) for x∉𝕁q, consider the auxiliary function:
(2.41)g(x)=f(x)-q2x={0if0≤x≤q-2,-(q2x-1)2q2-1ifq-2<x≤1.
Since [n-k]q/[n]q≤q-k for k=0,1,…,n, and [n-1]q/[n]q≥q-2 whenever qn≥q+1, it follows that, for sufficiently large n,
(2.42)Bn,q(g;x)=g([n-1]q[n]q)pn,n-1(q;x)+g(1)pnn(x).
Plain computations reveal
(2.43)g([n-1]q[n]q)=-(q-1)(qn-q-1)2(qn-1)2(q+1),
yielding
(2.44)Bn,q(g;x)=-(qn-q-1)2(qn-1)(q+1)xn-1(1-x)-(q2-1)xn.
Consequently, for x∉𝕁q, one obtains
(2.45)limn→∞Bn,q(g;x)={0if|x|<q-1,∞if|x|>q-1.
Since, by (1.10), Bn,q(f;x)=q2x+Bn,q(g;x), it follows that:
(2.46)limn→∞Bn,q(f;x)={q2xif|x|<q-1,∞if|x|>q-1,x≠1,q2+1q+1ifx=q-1,1ifx=1.
For x=-q-1, the limit does not exist. Additionally, it is not difficult to see that Bn,q(f;x)→f(x) as n→∞ uniformly on any compact set inside (-1/q2,1/q2), while on any interval outside of (-1/q2,1/q2), 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 Bn,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∈𝕁q and at some exemplary points x∉𝕁q are given in Table 2.Table 2
The values ofE(n,q,x)=Bn,q(f;x)-f(x) at some points x∈[0,1].
x
E
(
3,2
,
x
)
E
(
4,2
,
x
)
E
(
10,2
,
x
)
E
(
15,2
,
x
)
E
(
20,2
,
x
)
E
(
25,2
,
x
)
E
(
30,2
,
x
)
(
5
q
+
1
)
/
6
q
-
2.38
×
1
0
-
2
1.11
×
1
0
-
2
-
11.8
−268.
-
5.57
×
1
0
3
-
1.15
×
1
0
5
-
2.39
×
1
0
6
(
11
q
+
7
)
/
18
q
-
7.23
×
1
0
-
2
1.07
×
1
0
-
3
-
8.13
−101.
-
1.12
×
1
0
3
-
1.21
×
1
0
4
-
1.32
×
1
0
5
(
q
+
1
)
/
2
q
-
9.97
×
1
0
-
2
-
1.20
×
1
0
-
2
-
5.21
-
47.4
−368.
-
2.80
×
1
0
3
-
2.13
×
1
0
4
(
5
q
+
13
)
/
18
q
-
0.151
-
4.69
×
1
0
-
2
-
1.4
-
6.64
-
24.6
-
85.6
−293.
(
q
+
5
)
/
6
q
-
0.172
-
6.54
×
1
0
-
2
-
0.528
-
1.81
-
4.6
-
10.6
-
23.7
1
/
q
-
0.19
-
8.89
×
1
0
-
2
-
1.30
×
1
0
-
3
-
4.07
×
1
0
-
5
-
1.27
×
1
0
-
6
-
3.97
×
1
0
-
8
-
1.24
×
1
0
-
9
(
q
+
1
)
/
2
q
2
-
0.180
-
9.98
×
1
0
-
2
0.052
7.59
×
1
0
-
2
8.16
×
1
0
-
2
8.29
×
1
0
-
2
8.32
×
1
0
-
2
1
/
q
2
-
0.103
-
5.57
×
1
0
-
2
-
9.75
×
1
0
-
4
-
3.05
×
1
0
-
5
-
9.54
×
1
0
-
7
-
2.98
×
1
0
-
8
-
9.31
×
1
0
-
10
(
q
+
1
)
/
2
q
3
-
5.38
×
1
0
-
2
-
2.38
×
1
0
-
2
-
7.92
×
1
0
-
5
-
5.89
×
1
0
-
7
-
4.37
×
1
0
-
9
-
3.24
×
1
0
-
11
-
2.40
×
1
0
-
13
1
/
q
3
-
2.21
×
1
0
-
2
-
7.15
×
1
0
-
3
-
2.22
×
1
0
-
6
-
2.17
×
1
0
-
9
-
2.12
×
1
0
-
12
-
2.07
×
1
0
-
15
-
2.02
×
1
0
-
18
(
q
+
1
)
/
2
q
4
-
1.20
×
1
0
-
2
-
3.04
×
1
0
-
3
-
1.72
×
1
0
-
7
-
4.01
×
1
0
-
11
-
9.29
×
1
0
-
15
-
2.15
×
1
0
-
18
-
4.99
×
1
0
-
22
1
/
q
4
-
5.09
×
1
0
-
3
-
9.05
×
1
0
-
4
-
4.64
×
1
0
-
9
-
1.42
×
1
0
-
13
-
4.34
×
1
0
-
18
-
1.32
×
1
0
-
22
-
4.04
×
1
0
-
27
(
q
+
1
)
/
2
q
5
-
2.8
×
1
0
-
3
-
3.83
×
1
0
-
4
-
3.54
×
1
0
-
10
-
2.57
×
1
0
-
15
-
1.86
×
1
0
-
20
-
1.35
×
1
0
-
25
-
9.78
×
1
0
-
31
1
/
q
5
-
1.22
×
1
0
-
3
-
1.14
×
1
0
-
4
-
9.35
×
1
0
-
12
-
8.96
×
1
0
-
18
-
8.55
×
1
0
-
24
-
8.15
×
1
0
-
30
-
7.77
×
1
0
-
36
(
q
+
1
)
/
2
q
6
-
6.77
×
1
0
-
4
-
4.81
×
1
0
-
5
-
7.08
×
1
0
-
13
-
1.61
×
1
0
-
19
-
3.64
×
1
0
-
26
-
8.25
×
1
0
-
33
-
1.87
×
1
0
-
39
1
/
q
6
-
2.98
×
1
0
-
4
-
1.43
×
1
0
-
5
-
1.86
×
1
0
-
14
-
5.56
×
1
0
-
22
-
1.66
×
1
0
-
29
-
4.94
×
1
0
-
37
-
1.47
×
1
0
-
44
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
1
/
q
19
-
4.33
×
1
0
-
12
-
2.61
×
1
0
-
17
-
1.13
×
1
0
-
49
-
9.21
×
1
0
-
77
-
7.44
×
1
0
-
104
-
6.01
×
1
0
-
131
-
4.86
×
1
0
-
158
(
q
+
1
)
/
2
q
20
-
2.44
×
1
0
-
12
-
1.10
×
1
0
-
17
-
8.52
×
1
0
-
51
-
1.64
×
1
0
-
78
-
3.15
×
1
0
-
106
-
6.03
×
1
0
-
134
-
1.16
×
1
0
-
161
1
/
q
20
-
1.08
×
1
0
-
12
-
3.26
×
1
0
-
18
-
2.22
×
1
0
-
52
-
5.62
×
1
0
-
81
-
1.42
×
1
0
-
109
-
3.58
×
1
0
-
138
-
9.04
×
1
0
-
167
(
q
+
1
)
/
2
q
21
-
6.09
×
1
0
-
13
-
1.37
×
1
0
-
18
-
1.66
×
1
0
-
53
-
1.00
×
1
0
-
82
-
6.0
×
1
0
-
112
-
3.60
×
1
0
-
141
-
2.15
×
1
0
-
170
1
/
q
21
-
2.71
×
1
0
-
13
-
4.07
×
1
0
-
19
-
4.33
×
1
0
-
55
-
3.43
×
1
0
-
85
-
2.71
×
1
0
-
115
-
2.14
×
1
0
-
145
-
1.68
×
1
0
-
175
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
1
/
q
49
-
3.76
×
1
0
-
30
-
2.11
×
1
0
-
44
-
5.98
×
1
0
-
131
-
3.4
×
1
0
-
203
-
1.93
×
1
0
-
275
-
1.09
×
1
0
-
347
-
6.17
×
1
0
-
420
(
q
+
1
)
/
2
q
50
-
2.11
×
1
0
-
30
-
8.88
×
1
0
-
45
-
4.49
×
1
0
-
132
-
6.06
×
1
0
-
205
-
8.14
×
1
0
-
278
-
1.09
×
1
0
-
350
-
1.47
×
1
0
-
423
1
/
q
50
-
9.39
×
1
0
-
31
-
2.63
×
1
0
-
45
-
1.17
×
1
0
-
133
-
2.08
×
1
0
-
207
-
3.67
×
1
0
-
281
-
6.50
×
1
0
-
355
-
1.15
×
1
0
-
428Figure 3
Graphs ofy=f(x) andy=Bn,2(f;x),n=5,8.Remark 2.11.
Following Charalambides [9], consider a sequence of random variables {Xn(j)}n=1∞ possessing the distributions Pn(j) given by
(2.47)P{Xn(j)=[n-k]q[n]q}=pn,n-k(q;q-j),k=0,1,…,n.
Let I(q-j) denote a random variable with the δ-distribution concentrated at q-j. Theorem 2.1 implies that Xn(j)→I(q-j) in distribution.Generally speaking, Theorem2.1 shows that the q-Bernstein polynomials with q>1 possess an “interpolation-type” property on 𝕁q. Information on interpolation of functions with nodes on a geometric progression can be found in, for example, [18] by Schoenberg.


## 3. On theq-Bernstein Polynomials of the Weierstrass-Type Functions

In this section, theq-Bernstein polynomials of the functions with “bad” smoothness are considered. Let φ(x)∈C[-1,1] satisfy the condition:
(3.1)φ(0)>φ(x)forx∈[-1,1]∖{0}.
The letter φ will also denote a 2-periodic continuation of φ(x) on (-∞,∞).Definition 3.1.
Leta,b∈ℝ satisfy 0<a<1<ab. A function f(x) is said to be Weierstrass-type if
(3.2)f(x)=∑k=0∞akφ(bkx).
Notice that f(x) is continuous if and only if φ(-1)=φ(1). For φ(x)=cosπx 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.
Iff(x) is a Weierstrass-type function, then the sequence Bn,q(f;x) of its q-Bernstein polynomials is not uniformly bounded on any interval [0,c].Proof.
To prove the theorem, the following representation ofq-Bernstein polynomials (see [15], formulae (6) and (7)) is used:
(3.3)Bn,q(f;x)=∑k=0nλknf[0;1[n]q;…;[k]q[n]q]xk,
where
(3.4)λ0n=λ1n=1,λkn=(1-1[n]q)⋯(1-[k-1]q[n]q),k=2,…,n,
and f[x0;x1;…;xk] denote the divided differences of f, that is,
(3.5)f[x0]=f(x0),f[x0;x1]=f(x1)-f(x0)x1-x0,…,f[x0;x1;…;xk]=f[x1;…;xk]-f[x0;…;xk-1]xk-x0.
When q=1, the well-known representation for the classical Bernstein polynomials is recovered and the numbers λkn are the eigenvalues of the Bernstein operator, see [20], Chapter 4, Section 4.1 and [21]. The latter result has been extended to the case q≠1 in [15].
Clearly, it suffices to consider the case0<c<1. From (3.3), it follows that
(3.6)Bn,q′(f;0)=λ1nf[0;1[n]q]=[n]q{f(1[n]q)-f(0)},
and, hence,
(3.7)|Bn,q′(f;0)|=[n]q{f(0)-f(1[n]q)}=[n]q∑k=0∞ak{φ(0)-φ(bk[n]q)}.
What remains is to find a lower bound for |Bn,q′(f;0)|. Due to (3.1), all terms of the series are nonnegative and, therefore,
(3.8)|Bn,q′(f;0)|≥[n]qaj{φ(0)-φ(bj[n]q)}foranyj=0,1,…
Let j=jn be chosen in such a way that
(3.9)1b<bjn[n]q≤1.
For n>b, such a choice is possible because, in this case, inequality (3.9) implies that
(3.10)0<ln[n]qlnb-1<jn≤ln[n]qlnb.
Since the length of the interval (ln[n]q/lnb-1,ln[n]q/lnb] is 1, there is a positive integer, say, jn, such that jn∈(ln[n]q/lnb-1,ln[n]q/lnb]. The obvious inequality [n]q>qn-1 implies the following:
(3.11)jn≥(n-1)lnqlnb-1=:nlnqlnb-A,
with A=(lnq/lnb)+1 being a positive constant. Then, for n>b, it follows that
(3.12)|Bn,q′(f;0)|≥[n]qajnmint∈[1/b,1]{φ(0)-φ(t)}:=τ[n]qajn,
where τ>0 due to (3.1). Consequently,
(3.13)|Bn,q′(f;0)|≥τ[n]qbjn(ab)jn≥τ(ab)jn≥τ(ab)n(lnq/lnb)-A,
which leads to
(3.14)|Bn,q′(f;0)|≥Cρn,
where C=τ(ab)-A is a positive constant and ρ=(ab)(lnq/lnb)>1. Now, assume that {Bn,q(f;x)} is uniformly bounded on [0,c], that is, |Bn,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
(3.15)|Bn,q′(f;0)|≤2Mcn2∀n=1,2,…,
This proves the theorem because the latter estimate contradicts (3.14).To present an illustrative example, let us denote theNth partial sum of the series in (3.2) by hN, that is:
(3.16)hN(x)=∑k=0Nakφ(bkx).
Clearly, the function hN is an approximation of (3.2) satisfying the error estimate
(3.17)EN(x)=|f(x)-hN(x)|≤maxt∈[-1,1]|φ(t)|aN+11-a,∀x∈[0,1].Example 3.3.
Letφ(x)=(cosπx),a=1/2, and b=4. For N=20, one has E20(x)≤10-6. The graphs of h20(x) and the associated q-Bernstein polynomials Bn,q(h20;x) for q=2,  n=4,5, and 6 on the subintervals [0,0.55] and [0.55,1] are presented in Figures 4 and 5, respectively.Figure 4
Graphs ofy=h20(x) and y=Bn,2(f;x),  n=4,5,6.Figure 5
Graphs ofy=h20(x) and y=Bn,2(f;x),  n=4,5,6.



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*Source: 185948-2012-09-12.xml*