# On theq-Bernstein Polynomials of Unbounded  Functions with q>1

Authors: Sofiya Ostrovska; Ahmet Yaşar Özban
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The aim of this paper is to present new results related to theq-Bernstein polynomials Bn,q(f;x) of unbounded functions in the case q>1 and to illustrate those results using numerical examples. As a model, the behavior of polynomials Bn,q(f;x) is examined both theoretically and numerically in detail for functions on [0,1] satisfying f(x)~Kx−α as x→0+, where α>0 and K≠0 are real numbers.


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## Body

## 1. Introduction

In 1912, precisely a century ago, Bernstein [1] published his famous proof of the Weierstrass approximation theorem by introducing polynomials, known today as the Bernstein polynomials. Later, it was found that these polynomials possess many remarkable properties, which made them an area of intensive research with wide range of applications. See, for example [2, 3]. The importance of the Bernstein polynomials led to the discovery of their numerous generalizations aimed to provide appropriate tools for various areas of mathematics, such as approximation theory, computer-aided geometric design, and the statistical inference.Due to the speedy development of theq-calculus, recent generalizations based on the q-integers have emerged. A. Lupaş was the person who pioneered the work on the q-versions of the Bernstein polynomials. In 1987, he introduced (cf. [4]) a q-analogue of the Bernstein operator and investigated its approximation and shape-preserving properties. See also [5]. Subsequently, another generalization, called the q-Bernstein polynomials, was brought into the spotlight by Phillips [6] and was studied afterwards by a number of authors from different perspectives.To define these polynomials, let us recall some notions related to theq-calculus. See, for example, [7], Ch. 10. Let q>0. For any nonnegative integer k, the q-integer[k]q is defined by
(1)[k]q:=1+q+⋯+qk-1(k=1,2,…),[0]q:=0,
and the q-factorial[k]q! is defined by
(2)[k]q!:=[1]q[2]q⋯[k]q(k=1,2,…),[0]q!:=1.For integersk,n with 0≤k≤n, the q-binomial coefficient [nk]q is defined by
(3)[nk]q:=[n]g![k]q![n-k]q!.
We also use the following standard notations:
(4)(a;q)0≔1,(a;q)k≔∏s=0k-1(1-aqs),(a;q)∞:=∏s=0∞(1-aqs).Definition 1.
Letf:[0,1]→ℝ. The q-Bernstein polynomial of f is
(5)Bn,q(f;x):=∑k=0nf([k]q[n]q)pnk(q;x),n=1,2,…,
where the q-Bernstein basic polynomialspnk(q;x) are given by
(6)pnk(q;x):=[nk]qxk(x;q)n-k,k=0,1,…n.Polynomialspn0(q;x),pn1(q;x),…, and pnn(q;x) form the q-Bernstein basis in the linear space of the polynomials of degree at most n.Note that forq=1, Bn,q(f;x) is the classical Bernstein polynomial Bn(f;x). Conventionally, the name q-Bernstein polynomials is reserved for the case q≠1.Over the past years, theq-Bernstein polynomials have remained under intensive study, and new researches concerning not only the properties of the q-Bernstein polynomials, but also their various generalizations are constantly coming out (see, e.g., papers [8–14]). A detailed review of the results on the q-Bernstein polynomials along with the extensive bibliography has been provided in [10].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; see [15]. 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, its limit form called the q-deformed Poisson distribution defines the distribution of energy in a q-analogue of the coherent state; see [16].It has been known that theq-Bernstein polynomials inherit some of the properties of the classical Bernstein polynomials. For example, they possess the following endpoint interpolation property:
(7)Bn,q(f;0)=f(0),Bn,q(f;1)=f(1),n=1,2,…,q>0,
and have linear functions as their fixed points:
(8)Bn,q(ax+b;x)=ax+b,n=1,2,…,q>0.The latter follows from the identity:(9)∑k=0npnk(q;x)=1∀n=1,2,…,andallq>0.Nevertheless, 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. See, for example [10]. This lack of similarity stems from the fact that while for 0<q<1, the q-Bernstein operators given by
(10)Bn,q:f↦Bn,q(f;·)
are positive linear operators on C[0,1]; this is no longer valid for q>1. In addition, the case q>1 is aggravated by the rather irregular behavior of basic polynomials (6), which, in this case, combine the fast increase in magnitude with the sign oscillations. For details see [17], where it has been shown that the norm ∥Bn,q∥ increases rather rapidly in both n and q, namely,
(11)∥Bn,q∥~2e·q(n(n-1)/2)nasn⟶∞,q⟶+∞.In the present paper, theq-Bernstein polynomials in the case q>1 are studied for unbounded functions—a problem which has not been considered before. The approximation of unbounded functions by the classical Bernstein polynomials was investigated by Lorentz in [2] and, recently, by Weba in [18].Throughout the paper,q>1 is assumed to be fixed. In presenting the results, the notation 𝕁q is used for the time scale:
(12)𝕁q:={0}∪{q-j}j=0∞.First, we prove that for a certain class of unbounded functions on[0,1], their sequence of the q-Bernstein polynomials is approximating on 𝕁q. Further, the behavior of Bn,q(f;x) has been considered for functions f:[0,1]→ℝ which are continuous on (0,1] and satisfy
(13)f(x)~Kx-αasx⟶0+,
where α>0 and K∈ℝ∖{0}. Previously, the q-Bernstein polynomials with q>1 of the power functions xα, where α>0, were studied in [19]. There, it was proved that in the case α>0,Bn,q(xα;x)→xα uniformly on [0,1] if and only if α∈ℕ.Numerical examples are used both to illustrate Theorems3 and 6 and also to discuss the significance of the assumptions therein. All the numerical results have been calculated in a Maple 8 environment.


## 2. Theq-Bernstein Polynomials of  Unbounded Functions

It has been known that forq>1, all functions continuous on [0,1] are approximated by their q-Bernstein polynomials on the time scale 𝕁q. Here, it will be proved that this fact remains true for functions which are continuous from the left at all points of 𝕁q.Forf:[0,1]→ℝ,j∈ℤ+ and t∈[0,q-j], we set
(14)Ωf,q-j(t)=supx∈[q-j-t,q-j]|f(x)-f(q-j)|.Lemma 2.
Letf:[0,1]→ℝ be bounded on [q-j,1], where j∈ℕ, and letMj=supx∈[q-j,1]|f(x)|.Then, fornlarge enough, the following estimate holds:(15)|Bn,q(f;q-j)-f(q-j)|≤Ωf,q-j(1qn-1)+2Mj[j]qqn.Proof.
Forj=0, the statement is obvious due to the endpoint interpolation property (7). Therefore, it is assumed that j≥1.
First, it should be noticed thatpn,n-k(q;q-j)≥0 with pn,n-k(q;q-j)=0 for k>j, and that, by virtue of (9), ∑k=0jpn,n-k(q;q-j)=1. Then, for n>j, the following equality is true:
(16)Bn,q(f;q-j)=∑k=0jf([n-k]q[n]q)pn,n-k(q;q-j).
Then
(17)|Bn,q(f;q-j)-f(q-j)|≤∑k=0j|f([n-k]q[n]q)-f(q-j)|pn,n-k(q;q-j)≤|f([n-j]q[n]q)-f(q-j)|·pn,n-j(q;q-j)+2Mj·∑k=0j-1pn,n-k(q;q-j)
for n large enough to satisfy [n-j+1]q/[n]q>q-j. Since
(18)|[n-j]q[n]q-q-j|=qj-1qj(qn-1)≤1qn-1,pn,n-j(q;q-j)=(1-1qn)⋯(1-1qn-j+1)≤1,
the first term can be estimated as follows:
(19)|f([n-k]q[n]q)-f(q-j)|·pn,n-j(q;q-j)≤Ωf,q-j(1qn-1).
To estimate the second term, one can write
(20)∑k=0j-1pn,n-k(q;q-j)=1-pn,n-j(q;q-j)=1-(1-1qn)⋯(1-1qn-j+1).
By virtue of the inequality 1-(1-α1)⋯(1-αm)≤∑k=1mαk, which is satisfied for all α1,…,αm∈(0,1), it follows that
(21)∑k=0j-1pn,n-k(q;q-j)≤1qn∑k=0j-1qk=[j]qqn.With the help of Lemma2, the following result can be derived easily.Theorem 3.
Iff:[0,1]→ℝ is continuous from the left at x=q-j and bounded on [q-j,1], then Bn,q(f;q-j)→f(q-j) as n→∞.Proof.
The statement follows immediately from estimate (15). Indeed, limt→0+Ωf,q-j(t)=0 since f is continuous from the left at q-j, and limn→∞((2Mj[j]q)/(qn))=0 since Mj<∞.Corollary 4.
Iff:[0,1]→ℝ is continuous from the left at every q-j,  j∈ℕ, then Bn,q(f;q-j)→f(q-j) as n→∞ for all q-j∈𝕁q.Remark 5.
The conditions of the theorem are essential and cannot be left out entirely, as it will be shown by Example8. Furthermore, it is not difficult to see that if f(x) is bounded on [q-j,1], then the condition
(22)limn→∞f([n-j]q[n]q)=f(q-j)
is necessary for the approximation at x=q-j.The power functionsx-α,α>0 supply a natural family of functions discontinuous at 0 satisfying the conditions of Theorem 3 for all q-j∈𝕁q∖{0}. Therefore, it is interesting to consider the q-Bernstein polynomials of functions whose behavior at 0 is similar to that of the power functions. The next theorem investigates the behavior of the q-Bernstein polynomials of such functions on the set ℝ∖𝕁q.Theorem 6.
Letf:[0,1]→ℝ so that f(x)∈C(0,1] and limx→0+xαf(x)=K≠0. Then, for q≥2,
(23)Bn,q(f;x)⟶∞asn⟶∞∀x∈ℝ∖𝕁q.Proof.
We setu(x)=xαf(x),x∈(0,1] with u(0)=K. It can be readily seen that(24)Bn,q(f;x)=f(0)pn0(q;x)+∑k=1nu([k]q[n]q)[n]qα[k]qαpnk(q;x)=(-1)n[n]qαqn(n-1)/2xn·{f(0)[n]qα+∑k=1nu([k]q/[n]q)(-1)kqk(q-n;q)k[k]qα(qk-1)⋯(q-1)×(1x;1q)n-k}=(-1)n[n]qαqn(n-1)/2xn·{f(0)[n]qα+∑k=1∞‍ckn(q;x)},
where
(25)ckn(q;x)={u([k]q/[n]q)(-1)kqk(q-n;q)k[k]qα(qk-1)⋯(q-1)·(1x;1q)n-kifk≤n,0ifk>n.
Since, if x≠0,[n]qαqn(n-1)/2xn→∞ and (f(0)/[n]qα)→0 as n→∞, it suffices to prove that limn→∞∑k=1∞‍ckn(q;x)≠0 whenever x∉𝕁q. Let maxx∈[0,1]|u(x)|=M. Then, it is not difficult to see that
(26)|ckn(q;x)|≤Mqk[k]qα(qk-1)⋯(q-1)·(-1|x|;1q)∞=:dk(q;x)
and that by the ratio test, ∑k=1∞‍dk(q;x)<∞. Hence, by the Lebesgue dominated convergence theorem,
(27)limn→∞∑k=1∞‍ckn(q;x)=∑k=1∞‍(limn→∞ckn(q;x))=(1x;1q)∞·∑k=1∞‍K(-1)kqk[k]qα(qk-1)⋯(q-1)=:K(1x;1q)∞·∑k=1∞‍(-1)kak.
Obviously, K(1/x;1/q)∞≠0 if x∉𝕁q, and it is left to show that
(28)∑k=1∞‍(-1)kak≠0forq≥2.
Consider
(29)ak+1ak=qqk+1-1·[k]qα[k+1]qα≤qqk+1-1≤qq2-1<1ifq≥2.
Consequently,
(30)∑k=1∞‍(-1)kak∑=-(a1-a2)-(a3-a4)-⋯-(a2k-1-a2k)-⋯<0.
The grouping of terms is justified, because the series is absolutely convergent.
As a result, one concludes thatlimn→∞∑k=1∞‍ckn(q;x)≠0 when x∉𝕁q, and the proof is complete.


## 3. Numerical Examples

In this section, we provide the results of some numerical experiments to exemplify the validity of the theoretical results through a few test examples performed using high-precision computations with Maple 8. All of these computations are performed with 1000 digits so as to minimize the round-off errors, and 3 or 8 digits are used in showing the results in the tables.Example 7.
Letg(x) be defined by
(31)g(x)={1xifx∈(0,1]0ifx=0.
The graphs of y=g(x) and y=Bn,q(g;x) for q=2,  n=4 are exhibited in Figure 1. Similarly, Figure 2 represents the graphs of y=g(x) and y=Bn,q(g;x) for q=2,  n=5,6 over the subintervals [0,0.3], [0.3,0.5], and [0.5,1], respectively.Figure 1
Graphs ofy=g(x) and y=Bn,2(g;x),n=4.Graphs ofy=g(x) and y=Bn,2(g;x),  n=5,6.
(a)
(b)
(c)In addition, in Table1, the values of the error functionE(n,q,x):=Bn,q(g;x)-g(x) with q=2 at some points x∈[0,1] are presented. These points are taken both in 𝕁q and in [0,1]∖𝕁q. It can be observed from the table that for j<n, the values of the error function at the points of 𝕁q are close to 0, while, at the points in between, they become rather large in magnitude, as it can be noticed from the upper part of Table 1. On the other hand, if x≤q-j with j>n, then |E(n,q,x)|≥g(x)-g(1/[n]q)=1/x-[n]q. The bottom part of the table refers to this case. Finally, the middle part of the table shows the transition between the two cases.Table 1
The values ofE(n,q,x)=Bn,q(g;x)-g(x) for q=2 at some points x∈[0,1].
x
E
(
5
,
q
,
x
)
E
(
10
,
q
,
x
)
E
(
20
,
q
,
x
)
E
(
30
,
q
,
x
)
E
(
50
,
q
,
x
)
(
q
+
1
)
/
2
q
-
106.
4.50
×
1
0
12
3.61
×
1
0
56
3.68
×
1
0
130
7.77
×
1
0
368
1
/
q
9.7
×
1
0
-
3
2.78
×
1
0
-
3
2.79
×
1
0
-
7
2.73
×
1
0
-
10
2.6
×
1
0
-
16
(
q
+
1
)
/
2
q
2
6.25
-
7.34
×
1
0
9
-
5.73
×
1
0
50
-
5.71
×
1
0
121
-
1.15
×
1
0
354
1
/
q
2
4.28
×
1
0
-
2
1.14
×
1
0
-
3
1.19
×
1
0
-
6
1.16
×
1
0
-
9
1.1
×
1
0
-
15
(
q
+
1
)
/
2
q
3
-
1.0
3.13
×
1
0
7
2.37
×
1
0
45
2.3
×
1
0
113
4.43
×
1
0
339
1
/
q
3
0.154
1.55
×
1
0
-
3
3.91
×
1
0
-
6
3.82
×
1
0
-
9
3.64
×
1
0
-
15
⋮
⋮
⋮
⋮
⋮
⋮
1
/
q
19
-
724
.
E
(
5
,
q
,
x
)
+
6.20
×
1
0
-
2
109.
7.32
×
1
0
-
2
6.98
×
1
0
-
8
(
q
+
1
)
/
2
q
20
-
836
.
E
(
5
,
q
,
x
)
+
4.65
×
1
0
-
2
-
38.6
-
1.48
×
1
0
18
-
1.27
×
1
0
142
1
/
q
20
-
1.02
×
1
0
3
E
(
5
,
q
,
x
)
+
3.10
×
1
0
-
2
-
355.
0.207
1.98
×
1
0
-
7
⋮
⋮
⋮
⋮
⋮
⋮
1
/
q
49
-
2.37
×
1
0
7
E
(
5
,
q
,
x
)
+
5.78
×
1
0
-
11
E
(
5
,
q
,
x
)
+
1.91
×
1
0
-
6
E
(
5
,
q
,
x
)
+
6.25
×
1
0
-
2
3.58
×
1
0
6
(
q
+
1
)
/
2
q
50
-
2.74
×
1
0
7
E
(
5
,
q
,
x
)
+
4.33
×
1
0
-
11
E
(
5
,
q
,
x
)
+
1.43
×
1
0
-
6
E
(
5
,
q
,
x
)
+
4.69
×
1
0
-
2
-
1.26
×
1
0
6
1
/
q
50
-
3.36
×
1
0
7
E
(
5
,
q
,
x
)
+
2.89
×
1
0
-
11
E
(
5
,
q
,
x
)
+
9.54
×
1
0
-
7
E
(
5
,
q
,
x
)
+
3.12
×
1
0
-
2
-
1.16
×
1
0
7In general, if a functionf:[0,1]→ℝ does not satisfy the conditions of Theorem 3, then neither the approximation on 𝕁q nor the divergence of {Bn,q(f;x)} for all x∈ℝ∖𝕁q is guaranteed. This fact is illustrated by the following example.Example 8.
Consider the functionh defined by
(32)h(x)={q1-qx-q2q-1ifx∈[q-2,q-1)0otherwise.
Then, for n large enough,
(33)Bn,q(h;x)=h([n-1]q[n]q)pn,n-1(q;x)=q(qn-1)-q2q-1·qn-1q-1xn-1(1-x)~q3(1-x)(q-1)2·(q2x)n-1,n⟶∞.
Therefore,
(34)limn→∞Bn,q(h;x)={0if|x|<q-2,∞if|x|>q-2,x≠1,q(q+1)q-1ifx=q-2,0ifx=1.
Consequently, for the error function E(n,q,x), one has
(35)limn→∞E(n,q,x)={0ifx∈[0,q-2),+∞ifx∈(q-2,1),q(q+1)q-1ifx=q-2,0ifx=1.
If x=-q-2, the limit does not exist. Figure 3 exhibits the graphs of y=h(x) and y=Bn,q(h;x) for q=2, n=4, while Figure 4 provides those of y=h(x) and y=Bn,q(h;x) for q=2,  n=5,6 over the subintervals [0,0.6] and [0.6,1]. In addition, in Table 2, the values of the error function E(n,q,x) with q=2 at some points x∈[0,1] which are taken both in 𝕁q and in [0,1]∖𝕁q are presented.Table 2
The values ofE(n,q,x)=Bn,q(h;x)-h(x) for q=2 at some points x∈[0,1].
x
E
(
5
,
q
,
x
)
E
(
10
,
q
,
x
)
E
(
20
,
q
,
x
)
E
(
30
,
q
,
x
)
E
(
50
,
q
,
x
)
(
q
+
1
)
/
2
q
142.
3.92
×
1
0
4
2.32
×
1
0
9
1.37
×
1
0
14
4.79
×
1
0
23
1
/
q
56.2
2.04
×
1
0
3
2.10
×
1
0
6
2.15
×
1
0
9
2.25
×
1
0
15
(
q
+
1
)
/
2
q
2
18.2
187.
1.11
×
1
0
4
6.39
×
1
0
5
2.13
×
1
0
9
1
/
q
2
5.2675781
E
(
5
,
q
,
x
)
+
0.70900154
E
(
5
,
q
,
x
)
+
0.73239899
E
(
5
,
q
,
x
)
+
0.73242185
E
(
5
,
q
,
x
)
+
0.73242187
(
q
+
1
)
/
2
q
3
1.81
0.486
2.75
×
1
0
-
2
1.55
×
1
0
-
3
4.91
×
1
0
-
6
1
/
q
3
0.384
1.36
×
1
0
-
2
1.34
×
1
0
-
5
1.3
×
1
0
-
8
1.24
×
1
0
-
14
⋮
⋮
⋮
⋮
⋮
⋮
1
/
q
19
2.38
×
1
0
-
20
6.98
×
1
0
-
46
4.68
×
1
0
-
97
3.13
×
1
0
-
148
1.40
×
1
0
-
250
(
q
+
1
)
/
2
q
20
7.53
×
1
0
-
21
5.24
×
1
0
-
47
1.98
×
1
0
-
99
7.45
×
1
0
-
152
1.05
×
1
0
-
256
1
/
q
20
1.49
×
1
0
-
21
1.36
×
1
0
-
48
8.93
×
1
0
-
103
5.83
×
1
0
-
157
2.48
×
1
0
-
265
⋮
⋮
⋮
⋮
⋮
⋮
1
/
q
49
1.79
×
1
0
-
56
3.68
×
1
0
-
127
1.21
×
1
0
-
268
3.97
×
1
0
-
410
4.28
×
1
0
-
693
(
q
+
1
)
/
2
q
50
5.66
×
1
0
-
57
2.76
×
1
0
-
128
5.12
×
1
0
-
271
9.46
×
1
0
-
414
3.23
×
1
0
-
699
1
/
q
50
1.12
×
1
0
-
57
7.19
×
1
0
-
130
2.31
×
1
0
-
274
7.4
×
1
0
-
419
7.60
×
1
0
-
708Figure 3
Graphs ofy=h(x)  and y=Bn,2(h;x),  n=4.Graphs ofy=h(x)  and y=Bn,2(h;x),  n=5,6.
(a)
(b)It can be observed from the first three rows that asn increases, the values of the error function become very large—a trend which reflects its behavior on (q-2,1). At the point x=q-2, the values approach 6 from below, whereas for the remaining part of the table, the values of error function come close to zero as n increases, which is in full agreement with the limit given by (35).



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*Source: 349156-2013-03-31.xml*