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 {𝐵_𝑛,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 𝕁_𝑞={𝑞^−𝑗}^∞𝑗=0∪{0}, and that this result is sharp in the sense that the sequence {𝐵_𝑛,𝑞(𝑓;𝑥)}^∞𝑛=1 may be divergent for all 𝑥∈𝑅⧵𝕁_𝑞. 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 𝑓∶[0,1]→ℂ, 𝑞>0, and 𝑛∈ℕ. Then, the q-Bernstein polynomial of 𝑓 is defined by 𝐵_𝑛,𝑞(𝑓;𝑥)=^𝑛_𝑘=0𝑓[𝑘]^𝑞[𝑛]_𝑞𝑝^𝑛𝑘(𝑞;𝑥),(1.1) where 𝑝_𝑛𝑘(𝑞;𝑥)=⎡⎢⎢⎣𝑘𝑛⎤⎥⎥⎦_𝑞𝑥^𝑘(𝑥;𝑞)_𝑛−𝑘,𝑘=0,1,…𝑛,(1.2) with ^𝑛𝑘_𝑞 being the q-binomial coefficients given by ⎡⎢⎢⎣𝑘𝑛⎤⎥⎥⎦_𝑞=[𝑛]^𝑞![𝑘]_𝑞![𝑛−𝑘]_𝑞!,(1.3) and (𝑥;𝑞)_𝑚 being the 𝑞-Pochhammer symbol: (𝑥;𝑞)₀=1,(𝑥;𝑞)_𝑚=^𝑚−1_𝑠=0(1−𝑥𝑞^𝑠),(𝑥;𝑞)_∞=^∞_𝑠=0(1−𝑥𝑞^𝑠).(1.4) Here, for any nonnegative integer 𝑘, [𝑘]_𝑞!=[1]_𝑞[2]_𝑞⋯[𝑘]_𝑞(𝑘=1,2,…),[0]_𝑞!∶=1(1.5) are the q-factorials with [𝑘]_𝑞 being the q-integer given by [𝑘]_𝑞=1+𝑞+⋯+𝑞^𝑘−1(𝑘=1,2,…),[0]_𝑞∶=0.(1.6) 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): 𝐵_𝑛(𝑓;𝑥)=^𝑛_𝑘=0𝑓𝑘𝑛⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑥^𝑘(1−𝑥)^𝑛−𝑘,(1.7) conventionally, the name “𝑞-Bernstein polynomials” is reserved for the case 𝑞≠1.

Based on the 𝑞-Bernstein polynomials, the 𝑞-Bernstein operator on 𝐶[0,1] is given by 𝐵_𝑛,𝑞∶𝑓↦𝐵_𝑛,𝑞(𝑓;⋅).(1.8) 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: 𝐵_𝑛,𝑞(𝑓;0)=𝑓(0),𝐵_𝑛,𝑞(𝑓;1)=𝑓(1),𝑛=1,2,…,𝑞>0,(1.9) and leave the linear functions invariant: 𝐵_𝑛,𝑞(𝑎𝑡+𝑏;𝑥)=𝑎𝑥+𝑏,𝑛=1,2,…,𝑞>0.(1.10) In addition, the 𝑞-Bernstein basic polynomials (1.2) satisfy the identity ^𝑛_𝑘=0𝑝_𝑛𝑘(𝑞;𝑥)=1∀𝑛=1,2,…,∀𝑞>0.(1.11) 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≠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, ‖‖𝐵_𝑛,𝑞‖‖∼2𝑒⋅𝑞𝑛^{𝑛(𝑛−1)/2}as𝑛→∞,𝑞→+∞.(1.12) 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 𝕁_𝑞=𝑞^−𝑗^∞𝑗=0∪{0}(1.13) 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 𝕁_𝑞. Moreover, it is proved that 𝕁_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 𝕁_𝑞. 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 the 𝑞-Bernstein Polynomials on 𝕁_𝑞

In this paper, 𝑞>1 is considered fixed. It has been shown in [15], that, if a function 𝑓 is analytic in 𝐷_𝜀={𝑧∶|𝑧|<1+𝜀}, then it is uniformly approximated by its 𝑞-Bernstein polynomials on any compact set in 𝐷_𝜀, 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 𝕁_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 𝕁_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 ||𝐵_𝑛,𝑞𝑓;𝑞^−𝑗−𝑓𝑞^−𝑗||≤2𝜔_𝑓⎛⎜⎜⎝𝑞^−𝑗1−𝑞^−𝑗[𝑛]_𝑞⎞⎟⎟⎠,𝑗∈ℤ₊,(2.1) where 𝜔_𝑓 is the modulus of continuity of 𝑓 on [0,1].

Corollary 2.2. If 𝑗∈ℤ₊, then ||𝐵_𝑛,𝑞𝑓;𝑞^−𝑗−𝑓𝑞^−𝑗||≤2𝜔_𝑓2√1[𝑛]_𝑞,(2.2) that is, 𝐵_𝑛,𝑞(𝑓;𝑥) converges uniformly to 𝑓(𝑥) on the time scale 𝕁_𝑞.

Proof. The proof is rather straightforward. First, notice that 𝑝_𝑛𝑘(𝑞;𝑞^−𝑗)≥0 for all 𝑛,𝑘,𝑗, while ∑^𝑛𝑘=0𝑝_𝑛𝑘(𝑞;𝑞^−𝑗)=1 by virtue of (1.11). Then ||𝐵_𝑛,𝑞𝑓;𝑞^−𝑗−𝑓𝑞^−𝑗||≤^𝑛_𝑘=0||||𝑓[𝑘]_𝑞[𝑛]_𝑞−𝑓𝑞^−𝑗≤_𝑛||||𝑝_𝑛𝑘𝑞;𝑞^−𝑗_𝑘=0𝜔_𝑓||||[[𝑘𝑛]]^𝑞_𝑞−𝑞^−𝑗||||𝑝_𝑛𝑘≤𝜔_𝑓𝑞;𝑞^−𝑗(𝛿)^𝑛_𝑘=01+𝛿1²[[𝑘𝑛]]^𝑞_𝑞−𝑞^−𝑗²𝑝^𝑛𝑘𝑞;𝑞^−𝑗(2.3) for any 𝛿>0. Plain calculations (see, e.g., [13], formula (2.7)) show that 𝐵_𝑛,𝑞(𝑡−𝑥)²;𝑥=𝑥(1−𝑥)[𝑛]_𝑞,(2.4) which implies that ||𝐵_𝑛,𝑞𝑓;𝑞^−𝑗−𝑓𝑞^−𝑗||≤𝜔_𝑓(𝛿)⋅1+1𝛿²⋅𝑞^−𝑗1−𝑞^−𝑗[𝑛]_𝑞.(2.5) 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: ||𝐵_𝑛,𝑞𝑓;𝑞^−𝑗−𝑓𝑞^−𝑗||≤𝐶_𝑗(𝑞^−𝑛),𝑛→∞,where𝐶_𝑗→∞as𝑗→∞.(2.6) The condition 𝑓∈𝐶¹[0,1] cannot be left out completely, as the following example shows.

Example 2.4. Consider a function 𝑓∈𝐶[0,1] satisfying 𝑓(𝑥)=⎧⎪⎨⎪⎩0𝑞if𝑥∈0,𝑞⁻²∪𝑞⁻¹,1₋₁,−𝑥^𝛼if𝑥∈𝑞⁻²+𝑞⁻¹2,𝑞⁻¹,(2.7) where 0<𝛼<1. Then, for 𝑛 large enough, we have 𝐵_𝑛,𝑞𝑓;𝑞⁻¹−𝑓𝑞⁻¹=𝑓[𝑛−1]^𝑞[𝑛]_𝑞𝑝^{𝑛,𝑛−1}=𝑞−1𝑞;𝑞⁻¹𝑞^𝛼⋅(𝑞^𝑛−1)^1−𝛼𝑞^𝑛≥𝐶𝑞^−𝑛𝛼,(2.8) 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 𝕁_𝑞 may be rather unpredictable. The next theorem shows that the sequence {𝐵_𝑛,𝑞(𝑓;𝑥)} may be divergent for all 𝑥∈ℝ⧵𝕁_𝑞.

Theorem 2.5. Let 𝑓(𝑥)=𝑥^𝛼,  0<𝛼≤1/2. If 𝑞≥2, then 𝐵_𝑛,𝑞(𝑓;𝑥)→∞as𝑛→∞∀𝑥∈ℝ⧵𝕁_𝑞.(2.9)

Proof. The 𝑞-Bernstein polynomial of 𝑓 is 𝐵_𝑛,𝑞(𝑓;𝑥)=^𝑛_𝑘=0[[𝑘𝑛]]^𝑞_𝑞^𝛼𝑝_𝑛𝑘(𝑞;𝑥)=1[𝑛]^𝛼𝑞𝑛_𝑘=1[𝑘]^𝛼𝑞𝑝_𝑛𝑘(𝑞;𝑥).(2.10) Since for 𝑘=1,2,…,𝑛−1 one has 𝑝_𝑛𝑘(𝑞;𝑥)=(𝑞^𝑛−1)⋯𝑞^{𝑛−𝑘+1}−1𝑞^𝑘−1⋯(𝑞−1)𝑥^𝑘(𝑥;𝑞)_𝑛−𝑘=𝑞^{(2𝑛−𝑘+1)𝑘/2}(𝑞^−𝑛;𝑞)_𝑘𝑞^𝑘−1⋯(𝑞−1)(−1)^𝑛−𝑘𝑞^{(𝑛−𝑘)(𝑛−𝑘−1)/2}𝑥^𝑛𝑥1;1𝑞_𝑛−𝑘=(−1)^𝑛𝑞^{𝑛(𝑛−1)/2}𝑥^𝑛⋅(−1)^𝑘𝑞^𝑘(𝑞^−𝑛;𝑞)_𝑘𝑞^𝑘−1⋯(𝑞−1)⋅𝑥1;1𝑞_𝑛−𝑘,(2.11) it follows that 𝐵_𝑛,𝑞(𝑓;𝑥)=(−1)^𝑛𝑞^{𝑛(𝑛−1)/2}𝑥^𝑛[𝑛]^𝛼𝑞⋅𝑇(𝑛,𝑞,𝑥),(2.12) where 𝑇(𝑛;𝑞;𝑥)∶=(−1)^𝑛[𝑛]^𝛼𝑞𝑞^{𝑛(𝑛−1)/2}+^𝑛−1_𝑘=0(−1)^𝑘[𝑘]^𝛼𝑞𝑞^𝑘(𝑞^−𝑛;𝑞)_𝑘𝑞^𝑘−1⋯(𝑞−1)⋅𝑥1;1𝑞_𝑛−𝑘.(2.13) Obviously, lim_𝑛→∞𝑞^{𝑛(𝑛−1)/2}𝑥^𝑛[𝑛]^𝛼𝑞=∞forany𝑥≠0.(2.14) As such, the theorem will be proved if it is shown that lim_𝑛→∞𝑇(𝑛,𝑞,𝑥)≠0for𝑥∉𝕁_𝑞.(2.15) As lim_𝑛→∞(𝑞^{−𝑛(𝑛−1)/2}(−1)^𝑛[𝑛]^𝛼𝑞)=0, it suffices to prove that lim_{∞𝑛→∞}_𝑘=0𝑐_𝑘𝑛≠0when𝑥∉𝕁_𝑞,(2.16) where 𝑐_𝑘𝑛∶=⎧⎪⎨⎪⎩(−1)^𝑘[𝑘]^𝛼𝑞𝑞^𝑘(𝑞^−𝑛;𝑞)_𝑘𝑞^𝑘−1⋯(𝑞−1)⋅𝑥1;1𝑞_𝑛−𝑘0if𝑘≤𝑛−1,if𝑘≥𝑛.(2.17) The fact that (𝑞^−𝑛;𝑞)_𝑘≤1 and the inequality ||||1𝑥;1𝑞_𝑛−𝑘||||≤−1|𝑥|;1𝑞_𝑛−𝑘≤−1|𝑥|;1𝑞_∞(2.18) lead to ||𝑐_𝑘𝑛||≤𝑞^𝑘[𝑘]^𝛼𝑞𝑞^𝑘−1⋯(𝑞−1)−|𝑥1|;1𝑞_∞=∶𝑑_𝑘.(2.19) Now, since lim_𝑛→∞𝑐_𝑘𝑛=(−1)^𝑘𝑞^𝑘[𝑘]^𝛼𝑞𝑞^𝑘−1⋯(𝑞−1)𝑥1;1𝑞_∞,(2.20) and the series ∑^∞𝑘=0𝑑_𝑘 is convergent, the Lebesgues dominated convergence theorem implies lim_{∞𝑛→∞}_𝑘=0𝑐_𝑘𝑛=^∞_𝑘=0lim_𝑛→∞𝑐_𝑘𝑛=1(𝑞−1)^𝛼𝑥1;1𝑞_∞⋅^∞_𝑘=0(−1)^𝑘𝑎_𝑘,(2.21) where 𝑎_𝑘=𝑞^𝑘(𝑞^𝑘−1)^𝛼/(𝑞^𝑘−1)⋯(𝑞−1),𝑘=1,2,…. Moreover, 1(𝑞−1)^𝛼𝑥1;1𝑞_∞≠0whenever𝑥∉𝕁_𝑞.(2.22) How about the sum of the series in (2.21)? Consider the following two cases.
Case 1. 0<𝛼<1/3.
Let us show that 𝑎_𝑘+1<𝑎_𝑘,  𝑘=1,2,… for 𝑞≥2. Since 𝑎_𝑘+1𝑎_𝑘=𝑞𝑞^𝑘+1−1^1−𝛼𝑞^𝑘−1^𝛼,(2.23) for 𝑘≥2 it follows that 𝑎_𝑘+1𝑎_𝑘≤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.(2.24) Notice that (2.24) holds for any 𝛼∈(0,1). In addition, if 𝑘=1, then 𝑎₂𝑎₁=𝑞𝑞²−1^1−𝛼(𝑞−1)^𝛼=𝑞(𝑞−1)(𝑞+1)^1−𝛼.(2.25) The function in the r.h.s. is monotone decreasing in 𝑞, so 𝑎₂𝑎₁≤21⋅3^1−𝛼≤2₃√9<1.(2.26) Thus, {𝑎_𝑘}^∞𝑘=1 is a strictly decreasing sequence. Since all (𝑎_2𝑘−1−𝑎_2𝑘) are strictly positive, it follows that ^∞_𝑘=0(−1)^𝑘𝑎_𝑘=−𝑎₁−𝑎₂+𝑎₃−𝑎₄+⋯+𝑎_2𝑘−1−𝑎_2𝑘+⋯<0.(2.27)
Case 2. 1/3≤𝛼≤1/2.
Estimate (2.24) implies that ∑^∞𝑘=5(−1)^𝑘𝑎_𝑘<0. To prove the theorem, it suffices to show that 𝑎₁−𝑎₂+𝑎₃−𝑎₄>0 when 𝑞≥2. Denoting 𝑎_𝑖=(𝑞(𝑞−1)^𝛼/(𝑞−1))𝑔_𝑖(𝑞), 𝑖=1,2,3,4, we write the following: 𝑎₁−𝑎₂+𝑎₃−𝑎₄=𝑞(𝑞−1)^𝛼𝑞−1𝑔¹(𝑞)−𝑔₂(𝑞)+𝑔₃(𝑞)−𝑔₄(𝑞)=∶𝑞(𝑞−1)^𝛼𝑞−1𝐾(𝑞).(2.28) 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], 𝐾(𝑞)≥1−𝑔₁(2)+𝑔₂52−𝑔³(2)=1−23⋅3^𝛼+2002457349^𝛼−8315⋅15^𝛼=∶𝐿(𝛼).(2.29) The function 𝐿(𝛼) is strictly decreasing on [1/3,1/2]. Indeed, 𝐿^(𝛼)=−23⋅3^𝛼ln3+2002457349^𝛼ln394−3815⋅15^𝛼ln15(2.30) and, for 𝛼∈[1/3,1/2], 𝐿^(𝛼)≤−23⋅3^1/3ln3+2002457349^1/2ln394−3815⋅15^1/3ln15≤−0.4332<0,(2.31) whence 𝐿(𝛼)≥𝐿(1/2)≥1.096×10⁻³>0 for 𝛼∈[1/3,1/2].
Similarly, for 𝑞∈[5/2,3], 𝐾(𝑞)≥1−𝑔₁52+𝑔²(3)−𝑔₃5=1−10221⋅72^𝛼+9208⋅13^𝛼−80001496313⋅2803^𝛼=∶𝑀(𝛼).(2.32) 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, for 𝑞∈[3,+∞), we obtain 𝐾(𝑞)≥1−𝑔₁(3)−𝑔₃(3)=1−38⋅4^𝛼−2716640⋅40^𝛼=∶𝑁(𝛼).(2.33) Obviously, 𝑁(𝛼) is a strictly decreasing function for all 𝛼∈ℝ, whence, for 𝛼∈[1/3,1/2], 𝑁(𝛼)≥𝑁12≥0.239>0,(2.34) 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≥𝑞₀(𝛼).

An illustrative example is supplied below.

Example 2.7. Let 𝑓(𝑥)=√³𝑥. 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 2 represents 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 1 presents 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 𝕁_𝑞 and in [0,1]⧵𝕁_𝑞. It can be observed from Table 1 that, while at the points 𝑥∈𝕁_𝑞, the values of the error function are close to 0, at the points 𝑥∉𝕁_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].

Figure 1 

Graphs of 𝑦=𝑓(𝑥) and 𝑦=𝐵𝑛,2(𝑓;𝑥),  𝑛=4,5.

Figure 2 

Graphs of 𝑦=𝑓(𝑥) and 𝑦=𝐵𝑛,2(𝑓;𝑥),  𝑛=6,7.

Remark 2.8. Table 1 also shows that while the error function changes its sign for different values of 𝑥, for 𝑥=𝑞^−𝑗∈𝕁_𝑞, its values are negative, that is, 𝐵_𝑛,𝑞(𝑡^1/3;𝑞^−𝑗)<𝑓(𝑞^−𝑗) for 𝑞^−𝑗∈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 𝐵_𝑛,𝑞𝑓;𝑞^−𝑗≥𝑓𝑞^−𝑗correspondingly𝐵_𝑛,𝑞𝑓;𝑞^−𝑗≤𝑓𝑞^−𝑗,(2.35) for all 𝑞^−𝑗∈𝕁_𝑞.

Proof. It can be readily seen from (1.10) and (1.11) that ^𝑛_𝑘=0𝑝_𝑛𝑘𝑞;𝑞^−𝑗=1,^𝑛_𝑘=0[𝑘]_𝑞[𝑛]_𝑞𝑝_𝑛𝑘𝑞;𝑞^−𝑗=𝑞^−𝑗,(2.36) while 𝑝_𝑛𝑘(𝑞;𝑞^−𝑗)≥0. By virtue of Jensen's inequality, if 𝑓 is convex on [0,1], then whenever 𝑛∈ℕ and 𝑥₀,𝑥₁,…,𝑥_𝑛∈[𝑎,𝑏], there holds the following: ^𝑛_𝑘=0𝜆_𝑘𝑓𝑥_𝑘≥𝑓^𝑛_𝑘=0𝜆_𝑘𝑥_𝑘.(2.37) for all 𝜆₀,𝜆₁,…,𝜆_𝑛≥0 satisfying ∑^𝑛𝑘=0𝜆_𝑘=1. Setting 𝑥_𝑘=[𝑘]^𝑞[𝑛]_𝑞,𝜆_𝑘=𝑝_𝑛𝑘𝑞;𝑞^−𝑗,𝑘=0,1,…,𝑛,(2.38) and observing that ^𝑛_𝑘=0𝜆_𝑘𝑓𝑥_𝑘=𝐵_𝑛,𝑞𝑓;𝑞^−𝑗,(2.39) the required result is derived.

Example 2.10. Let 𝑓(𝑥)=⎧⎪⎨⎪⎩𝑞²𝑞₂𝑥if0≤𝑥≤𝑞⁻²,𝑥−𝑞²𝑥−1²𝑞²−1if𝑞⁻²<𝑥≤1.(2.40) The function is concave on [0,1] and, hence, according to the previous results, 𝐵_𝑛,𝑞(𝑓;𝑞^−𝑗)→𝑓(𝑞^−𝑗) as 𝑛→∞ from below for all 𝑗∈ℤ₊. To examine the behavior of polynomials 𝐵_𝑛,𝑞(𝑓;𝑥) for 𝑥∉𝕁_𝑞, consider the auxiliary function: 𝑔(𝑥)=𝑓(𝑥)−𝑞²𝑥=⎧⎪⎨⎪⎩0−𝑞₂if0≤𝑥≤𝑞⁻²,𝑥−1²𝑞²−1if𝑞⁻²<𝑥≤1.(2.41) Since [𝑛−𝑘]_𝑞/[𝑛]_𝑞≤𝑞^−𝑘 for 𝑘=0,1,…,𝑛, and [𝑛−1]_𝑞/[𝑛]_𝑞≥𝑞⁻² whenever 𝑞^𝑛≥𝑞+1, it follows that, for sufficiently large 𝑛, 𝐵_𝑛,𝑞(𝑔;𝑥)=𝑔[𝑛−1]^𝑞[𝑛]_𝑞𝑝^{𝑛,𝑛−1}(𝑞;𝑥)+𝑔(1)𝑝_𝑛𝑛(𝑥).(2.42) Plain computations reveal 𝑔[𝑛−1]^𝑞[𝑛]_𝑞=−(𝑞−1)(𝑞^𝑛−𝑞−1)²(𝑞^𝑛−1)²(𝑞+1),(2.43) yielding 𝐵_𝑛,𝑞(𝑔;𝑥)=−(𝑞^𝑛−𝑞−1)²(𝑞^𝑛−1)(𝑞+1)𝑥^𝑛−1(1−𝑥)−𝑞²−1𝑥^𝑛.(2.44) Consequently, for 𝑥∉𝕁_𝑞, one obtains lim_𝑛→∞𝐵_𝑛,𝑞(𝑔;𝑥)=⎧⎪⎨⎪⎩0∞if|𝑥|<𝑞⁻¹,if|𝑥|>𝑞⁻¹.(2.45) Since, by (1.10), 𝐵_𝑛,𝑞(𝑓;𝑥)=𝑞²𝑥+𝐵_𝑛,𝑞(𝑔;𝑥), it follows that: lim_𝑛→∞𝐵_𝑛,𝑞(𝑓;𝑥)=⎧⎪⎪⎨⎪⎪⎩𝑞²∞𝑥if|𝑥|<𝑞⁻¹,𝑞if|𝑥|>𝑞⁻¹,𝑥≠1,₂+1𝑞+11if𝑥=𝑞⁻¹,if𝑥=1.(2.46) For 𝑥=−𝑞⁻¹, the limit does not exist. Additionally, it is not difficult to see that 𝐵_𝑛,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 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∈𝕁_𝑞 and at some exemplary points 𝑥∉𝕁_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].

Figure 3 

Graphs of 𝑦=𝑓(𝑥) and𝑦=𝐵𝑛,2(𝑓;𝑥), 𝑛=5,8.

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 𝐏𝑋^𝑛(𝑗)=[𝑛−𝑘]^𝑞[𝑛]_𝑞=𝑝^{𝑛,𝑛−𝑘}𝑞;𝑞^−𝑗,𝑘=0,1,…,𝑛.(2.47) Let 𝐼(𝑞^−𝑗) denote a random variable with the 𝛿-distribution concentrated at 𝑞^−𝑗. Theorem 2.1 implies that 𝑋^𝑛(𝑗)→𝐼(𝑞^−𝑗) in distribution.

Generally speaking, Theorem 2.1 shows that the 𝑞-Bernstein polynomials with 𝑞>1 possess an “interpolation-type” property on 𝕁_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 𝜑(𝑥)∈𝐶[−1,1] satisfy the condition: 𝜑(0)>𝜑(𝑥)for𝑥∈[−1,1]⧵{0}.(3.1) The letter 𝜑 will also denote a 2-periodic continuation of 𝜑(𝑥) on (−∞,∞).

Definition 3.1. Let 𝑎,𝑏∈ℝ satisfy 0<𝑎<1<𝑎𝑏. A function 𝑓(𝑥) is said to be Weierstrass-type if 𝑓(𝑥)=^∞_𝑘=0𝑎^𝑘𝜑𝑏^𝑘𝑥.(3.2) 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. 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: 𝐵_𝑛,𝑞(𝑓;𝑥)=^𝑛_𝑘=0𝜆_𝑘𝑛𝑓0;1[𝑛]_𝑞;…;[𝑘]^𝑞[𝑛]_𝑞𝑥^𝑘,(3.3) where 𝜆_0𝑛=𝜆_1𝑛=1,𝜆_𝑘𝑛=1−1[𝑛]_𝑞⋯1−[𝑘[−𝑛]_𝑞1]^𝑞,𝑘=2,…,𝑛,(3.4) and 𝑓[𝑥₀;𝑥₁;…;𝑥_𝑘] denote the divided differences of 𝑓, that is, 𝑓𝑥₀=𝑓𝑥₀,𝑓𝑥₀;𝑥₁=𝑓𝑥¹−𝑓𝑥₀𝑥₁−𝑥₀𝑓𝑥,…,₀;𝑥₁;…;𝑥_𝑘=𝑓𝑥¹;…;𝑥_𝑘−𝑓𝑥₀;…;𝑥_𝑘−1𝑥_𝑘−𝑥₀.(3.5) 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≠1 in [15].
Clearly, it suffices to consider the case 0<𝑐<1. From (3.3), it follows that 𝐵^𝑛,𝑞(𝑓;0)=𝜆_1𝑛𝑓0;1[𝑛]_𝑞=[𝑛]^𝑞𝑓[𝑛1]_𝑞−𝑓(0),(3.6) and, hence, ||𝐵^𝑛,𝑞(𝑓;0)||=[𝑛]_𝑞𝑓(0)−𝑓[𝑛1]_𝑞=[𝑛]^𝑞∞_𝑘=0𝑎^𝑘𝜑(0)−𝜑[𝑏𝑛^𝑘]_𝑞.(3.7) What remains is to find a lower bound for |𝐵^𝑛,𝑞(𝑓;0)|. Due to (3.1), all terms of the series are nonnegative and, therefore, ||𝐵^𝑛,𝑞(𝑓;0)||≥[𝑛]_𝑞𝑎^𝑗𝜑(0)−𝜑[𝑛𝑏]^𝑗_𝑞forany𝑗=0,1,…(3.8) Let 𝑗=𝑗_𝑛 be chosen in such a way that 1𝑏<[𝑏𝑛]^𝑗𝑛_𝑞≤1.(3.9) For 𝑛>𝑏, such a choice is possible because, in this case, inequality (3.9) implies that 0<ln[𝑛]^𝑞ln𝑏−1<𝑗^𝑛≤ln[𝑛]^𝑞ln𝑏.(3.10) Since the length of the interval (ln[𝑛]_𝑞/ln𝑏−1,ln[𝑛]_𝑞/ln𝑏] is 1, there is a positive integer, say, 𝑗_𝑛, such that 𝑗_𝑛∈(ln[𝑛]_𝑞/ln𝑏−1,ln[𝑛]_𝑞/ln𝑏]. The obvious inequality [𝑛]_𝑞>𝑞^𝑛−1 implies the following: 𝑗_𝑛≥(𝑛−1)ln𝑞ln𝑏−1=∶𝑛llnn𝑞𝑏−𝐴,(3.11) with 𝐴=(ln𝑞/ln𝑏)+1 being a positive constant. Then, for 𝑛>𝑏, it follows that ||𝐵^𝑛,𝑞(𝑓;0)||≥[𝑛]_𝑞𝑎^𝑗𝑛min_{[]𝑡∈1/𝑏,1}{𝜑(0)−𝜑(𝑡)}∶=𝜏[𝑛]_𝑞𝑎^𝑗𝑛,(3.12) where 𝜏>0 due to (3.1). Consequently, ||𝐵^𝑛,𝑞(𝑓;0)||≥𝜏[𝑛]^𝑞𝑏^𝑗𝑛(𝑎𝑏)^𝑗𝑛≥𝜏(𝑎𝑏)^𝑗𝑛≥𝜏(𝑎𝑏)^{𝑛(lnq/ln𝑏)−𝐴},(3.13) which leads to ||𝐵^𝑛,𝑞(𝑓;0)||≥𝐶𝜌^𝑛,(3.14) where 𝐶=𝜏(𝑎𝑏)^−𝐴 is a positive constant and 𝜌=(𝑎𝑏)^(lnq/ln𝑏)>1. Now, assume that {𝐵_𝑛,𝑞(𝑓;𝑥)} is uniformly bounded on [0,𝑐], that is, |𝐵_𝑛,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 ||𝐵^𝑛,𝑞(𝑓;0)||≤2𝑀𝑐𝑛²∀𝑛=1,2,…,(3.15) This proves the theorem because the latter estimate contradicts (3.14).

To present an illustrative example, let us denote the 𝑁th partial sum of the series in (3.2) by ℎ_𝑁, that is: ℎ_𝑁(𝑥)=^𝑁_𝑘=0𝑎^𝑘𝜑𝑏^𝑘𝑥.(3.16) Clearly, the function ℎ_𝑁 is an approximation of (3.2) satisfying the error estimate 𝐸_𝑁(𝑥)=||𝑓(𝑥)−ℎ_𝑁(𝑥)||≤max_{[]𝑡∈−1,1}||𝜑(𝑡)||𝑎^𝑁+11−𝑎,∀𝑥∈[0,1].(3.17)

Example 3.3. Let 𝜑(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 𝐵_𝑛,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 of 𝑦=ℎ20(𝑥) and 𝑦=𝐵𝑛,2(𝑓;𝑥),  𝑛=4,5,6.

Figure 5 

Graphs of 𝑦=ℎ20(𝑥) and 𝑦=𝐵𝑛,2(𝑓;𝑥),  𝑛=4,5,6.

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.