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 .

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 .

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 .