Abstract and Applied Analysis

Volume 2012 (2012), Article ID 185948, 19 pages

doi:10.1155/2012/185948

Research Article

On the Sets of Convergence for Sequences of the -Bernstein Polynomials with

Sofiya Ostrovska and Ahmet Yaşar Özban

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

Received 27 March 2012; Accepted 19 June 2012

Academic Editor: Ngai-Ching Wong

Copyright © 2012 Sofiya Ostrovska and Ahmet Yaşar Özban. 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.

Abstract.

The aim of this paper is to present new results related to the convergence of the sequence of the -Bernstein polynomials in the case , where is a continuous function on . It is shown that the polynomials converge to uniformly on the time scale , and that this result is sharp in the sense that the sequence may be divergent for all . Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.

1. Introduction

Let , , and . Then, the q-Bernstein polynomial of is defined by

where

with being the q-binomial coefficients given by

and being the -Pochhammer symbol:

Here, for any nonnegative integer ,

are the q-factorials with being the q-integer given by

We use the notation from [[1], Ch. 10].

The polynomials , called the -Bernstein basic polynomials, form the -Bernstein basis in the linear space of polynomials of degree at most .

Although, for , the -Bernstein polynomial turns into the classical Bernstein polynomial :

conventionally, the name “-Bernstein polynomials” is reserved for the case .

Based on the -Bernstein polynomials, the -Bernstein operator on is given by

A detailed review of the results on the -Bernstein polynomials along with an extensive bibliography has been provided in [2]. In this field, new results concerning the properties of the -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 -Bernstein polynomials is attributed to the fact that they are closely related to the -binomial and the -deformed Poisson probability distributions (cf. [9]). The -binomial distribution plays an important role in the -boson theory, providing a -deformation for the quantum harmonic formalism. More specifically, it has been used to construct the binomial state for the -boson. Meanwhile, the -deformed Poisson distribution, which is the limit form of -binomial one, defines the energy distribution in a -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 -Bernstein polynomials retain some of the properties of the classical Bernstein polynomials. For example, they possess the end-point interpolation property:

and leave the linear functions invariant:

In addition, the -Bernstein basic polynomials (1.2) satisfy the identity

Furthermore, the -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 -Bernstein polynomials for are essentially different from those of the classical ones. What is more, the cases and 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 ,   are positive linear operators on , whereas for , no positivity occurs. In addition, the case 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 increases rather rapidly in both and . Namely,

This puts serious obstacles in the analysis of the convergence for . The challenge has inspired some papers by a number of authors dealing with the convergence of -Bernstein polynomials in the case (see, e.g., [7, 17]). However, there are still many open problems related to the behavior of the -Bernstein polynomials with (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 -Bernstein polynomials of continuous functions with , in the sense that every sequence converges uniformly on . Moreover, it is proved that 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 -Bernstein polynomials on the time scale . Section 3 is devoted to the -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 -Bernstein Polynomials on

In this paper, is considered fixed. It has been shown in [15], that, if a function is analytic in , then it is uniformly approximated by its -Bernstein polynomials on any compact set in , and, in particular, on .

In this study, attention is focused on the -Bernstein polynomials of “bad” functions, that is, functions which do not have an analytic continuation from to the unit disc. In general, such functions are not approximated by their -Bernstein polynomials on . Moreover, their -Bernstein polynomials may tend to infinity at some points of (a simple example has been provided in [15]). Here, it is proved that the divergence of may occur everywhere outside of , which is a “minimal” set of convergence.

However, in spite of this negative information, it will be shown that, for any , the sequence of its -Bernstein polynomials converges uniformly on the time scale .

The next statement generalizing Lemma 1 of [15] can be regarded as a discrete analogue of the Popoviciu Theorem.

Theorem 2.1. Let . Then

where is the modulus of continuity of on .

Corollary 2.2. If , then

that is, converges uniformly to on the time scale .

Proof. The proof is rather straightforward. First, notice that for all , while by virtue of (1.11). Then

for any . Plain calculations (see, e.g., [13], formula (2.7)) show that

which implies that

Then, one can immediately derive the result by choosing .

Remark 2.3. In [7], Wu has shown that if , then for any , one has:

The condition cannot be left out completely, as the following example shows.

Example 2.4. Consider a function satisfying

where . Then, for large enough, we have

where is a positive constant independent from .

As it has been already mentioned, the behavior of the -Bernstein polynomials in the case outside of the time scale may be rather unpredictable. The next theorem shows that the sequence may be divergent for all .

Theorem 2.5. Let ,  . If , then

Proof. The -Bernstein polynomial of is

Since for one has

it follows that

where

Obviously,

As such, the theorem will be proved if it is shown that

As , it suffices to prove that

where

The fact that and the inequality

lead to

Now, since

and the series is convergent, the Lebesgues dominated convergence theorem implies

where . Moreover,

How about the sum of the series in (2.21)? Consider the following two cases.

Case 1. .

Let us show that ,   for . Since

for it follows that

Notice that (2.24) holds for any . In addition, if , then

The function in the r.h.s. is monotone decreasing in , so

Thus, is a strictly decreasing sequence. Since all are strictly positive, it follows that

Case 2. .

Estimate (2.24) implies that . To prove the theorem, it suffices to show that when . Denoting , , we write the following:

We are left to show that is strictly positive for the specified values of and . First of all, notice that , while , and are strictly decreasing in on . Hence, for ,

The function is strictly decreasing on . Indeed,

and, for ,

whence for .

Similarly, for ,

Applying the same reasoning as done for , it can be shown that is strictly decreasing on . Since , it follows that for all .

Finally, for , we obtain

Obviously, is a strictly decreasing function for all , whence, for ,

which completes the proof.

Remark 2.6. It can be seen from the proof that, the statement of the theorem is true for any and .

An illustrative example is supplied below.

Example 2.7. Let . The graphs of and for and are exhibited in Figure 1. Similarly, Figure 2 represents the graphs of and for and over the subintervals and , respectively. In addition, Table 1 presents the values of the error function with at some points . The points are taken both in and in . It can be observed from Table 1 that, while at the points , the values of the error function are close to 0, at the points , the values of the error function may be very large in magnitude.

Table 1: The values of at some points .

−465.

Figure 1: Graphs of and ,  .

Figure 2: Graphs of and ,  .

Remark 2.8. Table 1 also shows that while the error function changes its sign for different values of , for , its values are negative, that is, for . This is a particular case of the following statement.

Theorem 2.9. Let . If is convex (concave) on , then

for all .

Proof. It can be readily seen from (1.10) and (1.11) that

while . By virtue of Jensen's inequality, if is convex on , then whenever and , there holds the following:

for all satisfying . Setting

and observing that

the required result is derived.

Example 2.10. Let

The function is concave on and, hence, according to the previous results, as from below for all . To examine the behavior of polynomials for , consider the auxiliary function:

Since for , and whenever , it follows that, for sufficiently large ,

Plain computations reveal

yielding

Consequently, for , one obtains

Since, by (1.10), , it follows that:

For , the limit does not exist. Additionally, it is not difficult to see that as uniformly on any compact set inside , while on any interval outside of , the function is not approximated by its -Bernstein polynomials. This agrees with the result from [17], Theorem 2.3. The graphs of and for ,   and on are given in Figure 3. The values of the error function at some points and at some exemplary points are given in Table 2.

Table 2: The values of at some points .

−268.

−101.

−368.

−293.

Figure 3: Graphs of and, .

Remark 2.11. Following Charalambides [9], consider a sequence of random variables possessing the distributions given by

Let denote a random variable with the -distribution concentrated at . Theorem 2.1 implies that in distribution.

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

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

In this section, the -Bernstein polynomials of the functions with “bad” smoothness are considered. Let satisfy the condition:

The letter will also denote a 2-periodic continuation of on .

Definition 3.1. Let satisfy . A function is said to be Weierstrass-type if

Notice that is continuous if and only if . For and a special choice of and (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 , 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 is a Weierstrass-type function, then the sequence of its -Bernstein polynomials is not uniformly bounded on any interval .

Proof. To prove the theorem, the following representation of -Bernstein polynomials (see [15], formulae (6) and (7)) is used:

where

and denote the divided differences of , that is,

When , the well-known representation for the classical Bernstein polynomials is recovered and the numbers 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 in [15].

Clearly, it suffices to consider the case . From (3.3), it follows that

and, hence,

What remains is to find a lower bound for . Due to (3.1), all terms of the series are nonnegative and, therefore,

Let be chosen in such a way that

For , such a choice is possible because, in this case, inequality (3.9) implies that

Since the length of the interval is 1, there is a positive integer, say, , such that . The obvious inequality implies the following:

with being a positive constant. Then, for , it follows that

where due to (3.1). Consequently,

which leads to

where is a positive constant and . Now, assume that is uniformly bounded on , that is, for all . By Markov's Inequality (cf., e.g., [22], Chapter 4, Section 1, pp. 97-98) it follows that

This proves the theorem because the latter estimate contradicts (3.14).

To present an illustrative example, let us denote the th partial sum of the series in (3.2) by , that is:

Clearly, the function is an approximation of (3.2) satisfying the error estimate

Example 3.3. Let , and . For , one has . The graphs of and the associated -Bernstein polynomials for ,  , and on the subintervals and are presented in Figures 4 and 5, respectively.

Figure 4: Graphs of and ,  .

Figure 5: Graphs of and ,  .

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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