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

Abstract

The aim of this paper is to present new results related to the convergence of the sequence of the q -Bernstein polynomials {[subscript] B n ,q[/subscript] (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 [subscript] ... q[/subscript] = {[superscript] q -j[/superscript] [superscript] } j =0 ∞[/superscript] ∪ {0 } , and that this result is sharp in the sense that the sequence {[subscript] B n ,q[/subscript] (f ;x )[superscript] } n =1 ∞[/superscript] may be divergent for all x ∈R \[subscript] ... q[/subscript] . Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Ngai-Ching Wong

1, Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

Received 27 March 2012; Accepted 19 June 2012

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.

1. Introduction

Let f : [ 0,1 ] [arrow right] ... , q >0 , and n ∈ ... . Then, the q-Bernstein polynomial of f is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with _{[ n k ] q} being the q-binomial coefficients given by [figure omitted; refer to PDF] and _{( x ;q ) m} being the q -Pochhammer symbol: [figure omitted; refer to PDF] Here, for any nonnegative integer k , [figure omitted; refer to PDF] are the q-factorials with _{[ k ] q} being the q-integer given by [figure omitted; refer to PDF] 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 ) : [figure omitted; refer to PDF] conventionally, the name " q -Bernstein polynomials" is reserved for the case q ...0;1 .

Based on the q -Bernstein polynomials, the q -Bernstein operator on C [ 0,1 ] is given by [figure omitted; refer to PDF] 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: [figure omitted; refer to PDF] and leave the linear functions invariant: [figure omitted; refer to PDF] In addition, the q - Bernstein basic polynomials ( 1.2) satisfy the identity [figure omitted; refer to PDF] 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 ...0;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, [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 _{... q} . Moreover, it is proved that _{... 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 _{... q} . Section 3is 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 the q -Bernstein Polynomials on _{... q}

In this paper, q >1 is considered fixed. It has been shown in [ 15], that, if a function f is analytic in _{D [straight epsilon]} = { z : | z | <1 + [straight epsilon] } , then it is uniformly approximated by its q -Bernstein polynomials on any compact set in _{D [straight epsilon]} , 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 _{... 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 _{... q} .

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 [figure omitted; refer to PDF] where _{ω f} is the modulus of continuity of f on [ 0,1 ] .

Corollary 2.2.

If j ∈ _{... +} , then [figure omitted; refer to PDF] that is, _{B n ,q} ( f ;x ) converges uniformly to f ( x ) on the time scale _{... q} .

Proof.

The proof is rather straightforward. First, notice that _{p nk} ( q ; ^{q -j} ) ...5;0 for all n ,k ,j , while ^{∑ k =0 n} ... _{p nk} ( q ; ^{q -j} ) =1 by virtue of ( 1.11). Then [figure omitted; refer to PDF] for any δ >0 . Plain calculations (see, e.g., [ 13], formula (2.7)) show that [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] 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 ∈ ^{C 1} [ 0,1 ] , then for any j ∈ _{... +} , one has: [figure omitted; refer to PDF] The condition f ∈ ^{C 1} [ 0,1 ] cannot be left out completely, as the following example shows.

Example 2.4.

Consider a function f ∈C [ 0,1 ] satisfying [figure omitted; refer to PDF] where 0 < α <1 . Then, for n large enough, we have [figure omitted; refer to PDF] 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 _{... q} may be rather unpredictable. The next theorem shows that the sequence { _{B n ,q} ( f ;x ) } may be divergent for all x ∈ ... \ _{... q} .

Theorem 2.5.

Let f ( x ) = ^{x α} , 0 < α ...4; 1 / 2 . If q ...5;2 , then [figure omitted; refer to PDF]

Proof.

The q -Bernstein polynomial of f is [figure omitted; refer to PDF] Since for k =1,2 , ... ,n -1 one has [figure omitted; refer to PDF] it follows that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Obviously, [figure omitted; refer to PDF] As such, the theorem will be proved if it is shown that [figure omitted; refer to PDF] As _{lim n [arrow right] ∞} ( ^{q -n ( n -1 ) /2 ( -1 ) n [ n ] q α} ) =0 , it suffices to prove that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The fact that _{( ^{q -n} ;q ) k} ...4;1 and the inequality [figure omitted; refer to PDF] lead to [figure omitted; refer to PDF] Now, since [figure omitted; refer to PDF] and the series ^{∑ k =0 ∞} ... _{d k} is convergent, the Lebesgues dominated convergence theorem implies [figure omitted; refer to PDF] where _{a k} = ^{q k ( q k -1 ) α} / ( ^{q k} -1 ) ... ( q -1 ) ,k =1,2 , ... . Moreover, [figure omitted; refer to PDF] How about the sum of the series in ( 2.21)? Consider the following two cases.

Case1 . 0 < α <1 /3 .

Let us show that _{a k +1} < _{a k} , k =1,2 , ... for q ...5;2 . Since [figure omitted; refer to PDF] for k ...5;2 it follows that [figure omitted; refer to PDF] Notice that ( 2.24) holds for any α ∈ ( 0,1 ) . In addition, if k =1 , then [figure omitted; refer to PDF] The function in the r.h.s. is monotone decreasing in q , so [figure omitted; refer to PDF] Thus, ^{{ _{a k} } k =1 ∞} is a strictly decreasing sequence. Since all ( _{a 2k -1} - _{a 2k} ) are strictly positive, it follows that [figure omitted; refer to PDF]

Case2 . 1 /3 ...4; α ...4;1 /2 .

Estimate ( 2.24) implies that ^{∑ k =5 ∞} ... ^{( -1 ) k} _{a k} <0 . To prove the theorem, it suffices to show that _{a 1} - _{a 2} + _{a 3} - _{a 4} >0 when q ...5;2 . Denoting _{a i} = ( q ^{( q -1 ) α} / ( q -1 ) ) _{g i} ( q ) , i =1,2 ,3,4 , we write the following: [figure omitted; refer to PDF] 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 ] , [figure omitted; refer to PDF] The function L ( α ) is strictly decreasing on [ 1 /3,1 /2 ] . Indeed, [figure omitted; refer to PDF] and, for α ∈ [ 1 /3,1 /2 ] , [figure omitted; refer to PDF] whence L ( α ) ...5;L ( 1 / 2 ) ...5;1.096 ×1 ^{0 -3} >0 for α ∈ [ 1 /3,1 /2 ] .

Similarly, for q ∈ [ 5 /2,3 ] , [figure omitted; refer to PDF] 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 ) ...5;0.238 >0 , it follows that M ( α ) >0 for all α ∈ [ 1 /3,1 /2 ] .

Finally, for q ∈ [ 3 , + ∞ ) , we obtain [figure omitted; refer to PDF] Obviously, N ( α ) is a strictly decreasing function for all α ∈ ... , whence, for α ∈ [ 1 /3,1 /2 ] , [figure omitted; refer to PDF] which completes the proof.

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 ...5; _{q 0} ( α ) .

An illustrative example is supplied below.

Example 2.7.

Let f ( x ) =x3 . 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 2represents 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 1presents 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 _{... q} and in [ 0,1 ] \ _{... q} . It can be observed from Table 1that, while at the points x ∈ _{... q} , the values of the error function are close to 0, at the points x ∉ _{... q} , the values of the error function may be very large in magnitude.

Table 1: The values of E (n ,q ,x ) = _{B n ,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 ) /2q	-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
[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]
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
[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]
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 -6

Figure 1: Graphs of y =f ( x ) and y = _{B n ,2} ( f ;x ) , n =4,5 .

[figure omitted; refer to PDF]

Figure 2: Graphs of y =f ( x ) and y = _{B n ,2} ( f ;x ) , n =6,7 .

[figure omitted; refer to PDF]

Remark 2.8.

Table 1also shows that while the error function changes its sign for different values of x , for x = ^{q -j} ∈ _{... q} , its values are negative, that is, _{B n ,q} ( ^{t 1 /3} ; ^{q -j} ) <f ( ^{q -j} ) for ^{q -j} ∈ _{... q} . This is a particular case of the following statement.

Theorem 2.9.

Let q >1 . If f ( x ) is convex (concave) on [ 0,1 ] , then [figure omitted; refer to PDF] for all ^{q -j} ∈ _{... q} .

Proof.

It can be readily seen from ( 1.10) and ( 1.11) that [figure omitted; refer to PDF] while _{p nk} ( q ; ^{q -j} ) ...5;0 . By virtue of Jensen's inequality, if f is convex on [ 0,1 ] , then whenever n ∈ ... and _{x 0} , _{x 1} , ... , _{x n} ∈ [ a ,b ] , there holds the following: [figure omitted; refer to PDF] for all _{λ 0} , _{λ 1} , ... , _{λ n} ...5;0 satisfying ^{∑ k =0 n} ... _{λ k} =1 . Setting [figure omitted; refer to PDF] and observing that [figure omitted; refer to PDF] the required result is derived.

Example 2.10.

Let [figure omitted; refer to PDF] The function is concave on [ 0,1 ] and, hence, according to the previous results, _{B n ,q} ( f ; ^{q -j} ) [arrow right]f ( ^{q -j} ) as n [arrow right] ∞ from below for all j ∈ _{... +} . To examine the behavior of polynomials _{B n ,q} (f ;x ) for x ∉ _{... q} , consider the auxiliary function: [figure omitted; refer to PDF] Since _{[ n -k ] q} / _{[ n ] q} ...4; ^{q -k} for k =0,1 , ... ,n , and _{[ n -1 ] q} / _{[ n ] q} ...5; ^{q -2} whenever ^{q n} ...5;q +1 , it follows that, for sufficiently large n , [figure omitted; refer to PDF] Plain computations reveal [figure omitted; refer to PDF] yielding [figure omitted; refer to PDF] Consequently, for x ∉ _{... q} , one obtains [figure omitted; refer to PDF] Since, by ( 1.10), _{B n ,q} (f ;x ) = ^{q 2} x + _{B n ,q} (g ;x ) , it follows that: [figure omitted; refer to PDF] For x = - ^{q -1} , the limit does not exist. Additionally, it is not difficult to see that _{B n ,q} ( f ;x ) [arrow right]f ( x ) as n [arrow right] ∞ 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 ∈ _{... q} and at some exemplary points x ∉ _{... q} are given in Table 2.

Table 2: The values of E (n ,q ,x ) = _{B n ,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 )
(5q +1 ) /6q	-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
(11q +7 ) /18q	-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 ) /2q	-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
(5q +13 ) /18q	-0.151	-4.69 ×1 0 -2	-1.4	-6.64	-24.6	-85.6	-293.
(q +5 ) /6q	-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
[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]
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
[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]	[vertical ellipsis]
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 -428

Figure 3: Graphs of y =f (x ) and y = _{B n ,2} ( f ;x ) , n =5 ,8 .

[figure omitted; refer to PDF]

Remark 2.11.

Following Charalambides [ 9], consider a sequence of random variables ^{{ X n ( j ) } n =1 ∞} possessing the distributions ^{P n ( j )} given by [figure omitted; refer to PDF] Let I ( ^{q -j} ) denote a random variable with the δ -distribution concentrated at ^{q -j} . Theorem 2.1implies that ^{X n (j )} [arrow right]I ( ^{q -j} ) in distribution.

Generally speaking, Theorem 2.1shows that the q -Bernstein polynomials with q >1 possess an "interpolation-type" property on _{... q} . Information on interpolation of functions with nodes on a geometric progression can be found in, for example, [ 18] by Schoenberg.

3. 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 [straight phi] ( x ) ∈C [ -1,1 ] satisfy the condition: [figure omitted; refer to PDF] The letter [straight phi] will also denote a 2-periodic continuation of [straight phi] ( x ) on ( - ∞ , ∞ ) .

Definition 3.1.

Let a ,b ∈ ... satisfy 0 <a <1 <ab . A function f ( x ) is said to be Weierstrass-type if [figure omitted; refer to PDF] Notice that f ( x ) is continuous if and only if [straight phi] ( -1 ) = [straight phi] ( 1 ) . For [straight phi] ( 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 [straight phi] ( 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 ] .

Proof.

To prove the theorem, the following representation of q -Bernstein polynomials (see [ 15], formulae (6) and (7)) is used: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and f [ _{x 0} ; _{x 1} ; ... ; _{x k} ] denote the divided differences of f , that is, [figure omitted; refer to PDF] When q =1 , the well-known representation for the classical Bernstein polynomials is recovered and the numbers _{λ kn} 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 ...0;1 in [ 15].

Clearly, it suffices to consider the case 0 <c <1 . From ( 3.3), it follows that [figure omitted; refer to PDF] and, hence, [figure omitted; refer to PDF] What remains is to find a lower bound for | ^{B n ,q [variant prime]} ( f ;0 ) | . Due to ( 3.1), all terms of the series are nonnegative and, therefore, [figure omitted; refer to PDF] Let j = _{j n} be chosen in such a way that [figure omitted; refer to PDF] For n >b , such a choice is possible because, in this case, inequality ( 3.9) implies that [figure omitted; refer to PDF] 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: [figure omitted; refer to PDF] with A = ( ln q / ln b ) +1 being a positive constant. Then, for n >b , it follows that [figure omitted; refer to PDF] where τ >0 due to ( 3.1). Consequently, [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] 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 ) | ...4;M for all x ∈ [ 0 ,c ] . By Markov's Inequality (cf., e.g., [ 22], Chapter 4, Section 1, pp. 97-98) it follows that [figure omitted; refer to PDF] This proves the theorem because the latter estimate contradicts ( 3.14).

To present an illustrative example, let us denote the N th partial sum of the series in ( 3.2) by _{h N} , that is: [figure omitted; refer to PDF] Clearly, the function _{h N} is an approximation of ( 3.2) satisfying the error estimate [figure omitted; refer to PDF]

Example 3.3.

Let [straight phi] ( x ) = ( cos πx ) ,a = 1 / 2 , and b =4 . For N =20 , one has _{E 20} ( x ) ...4;1 ^{0 -6} . The graphs of _{h 20} ( x ) and the associated q -Bernstein polynomials _{B n ,q} ( _{h 20} ;x ) for q =2 , n =4 ,5 , and 6 on the subintervals [ 0,0.55 ] and [ 0.55,1 ] are presented in Figures 4and 5, respectively.

Figure 4: Graphs of y = _{h 20} ( x ) and y = _{B n ,2} ( f ;x ) , n =4,5 ,6 .

[figure omitted; refer to PDF]

Figure 5: Graphs of y = _{h 20} ( x ) and y = _{B n ,2} ( f ;x ) , n =4,5 ,6 .

[figure omitted; refer to PDF]

Acknowledgment

The authors would like to express their sincere gratitude to Mr. P. Danesh from Atilim University Academic Writing and Advisory Centre for his help in the preparation of the paper.

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