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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 185948, 19 pages
Research Article

On the Sets of Convergence for Sequences of the π‘ž-Bernstein Polynomials with π‘ž>1

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

Received 27 March 2012; Accepted 19 June 2012

Academic Editor: Ngai-ChingΒ Wong

Copyright Β© 2012 Sofiya Ostrovska and Ahmet Yaşar Γ–zban. 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.


The aim of this paper is to present new results related to the convergence of the sequence of the π‘ž-Bernstein polynomials {𝐡𝑛,π‘ž(𝑓;π‘₯)} in the case π‘ž>1, where 𝑓 is a continuous function on [0,1]. It is shown that the polynomials converge to 𝑓 uniformly on the time scale π•π‘ž={π‘žβˆ’π‘—}βˆžπ‘—=0βˆͺ{0}, and that this result is sharp in the sense that the sequence {𝐡𝑛,π‘ž(𝑓;π‘₯)}βˆžπ‘›=1 may be divergent for all π‘₯βˆˆπ‘…β§΅π•π‘ž. 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 π‘“βˆΆ[0,1]β†’β„‚, π‘ž>0, and π‘›βˆˆβ„•. Then, the q-Bernstein polynomial of 𝑓 is defined by 𝐡𝑛,π‘ž(𝑓;π‘₯)=π‘›ξ“π‘˜=0𝑓[π‘˜]π‘ž[𝑛]π‘žξ‚Άπ‘π‘›π‘˜(π‘ž;π‘₯),(1.1) where π‘π‘›π‘˜βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯⎦(π‘ž;π‘₯)=π‘žπ‘₯π‘˜(π‘₯;π‘ž)π‘›βˆ’π‘˜,π‘˜=0,1,…𝑛,(1.2) with ξ€Ίπ‘›π‘˜ξ€»π‘ž being the q-binomial coefficients given by βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž=[𝑛]π‘ž![π‘˜]π‘ž![]π‘›βˆ’π‘˜π‘ž!,(1.3) and (π‘₯;π‘ž)π‘š being the π‘ž-Pochhammer symbol: (π‘₯;π‘ž)0=1,(π‘₯;π‘ž)π‘š=π‘šβˆ’1𝑠=0(1βˆ’π‘₯π‘žπ‘ ),(π‘₯;π‘ž)∞=βˆžξ‘π‘ =0(1βˆ’π‘₯π‘žπ‘ ).(1.4) Here, for any nonnegative integer π‘˜, [π‘˜]π‘ž[1]!=π‘ž[2]π‘žβ‹―[π‘˜]π‘ž[0](π‘˜=1,2,…),π‘ž!∢=1(1.5) are the q-factorials with [π‘˜]π‘ž being the q-integer given by [π‘˜]π‘ž=1+π‘ž+β‹―+π‘žπ‘˜βˆ’1[0](π‘˜=1,2,…),π‘žβˆΆ=0.(1.6) We use the notation from [[1], Ch. 10].

The polynomials 𝑝𝑛0(π‘ž;π‘₯),𝑝𝑛1(π‘ž;π‘₯),…,𝑝𝑛𝑛(π‘ž;π‘₯), called the π‘ž-Bernstein basic polynomials, form the π‘ž-Bernstein basis in the linear space of polynomials of degree at most 𝑛.

Although, for π‘ž=1, the π‘ž-Bernstein polynomial 𝐡𝑛,π‘ž(𝑓;π‘₯) turns into the classical Bernstein polynomial 𝐡𝑛(𝑓;π‘₯): 𝐡𝑛(𝑓;π‘₯)=π‘›ξ“π‘˜=0π‘“ξ‚€π‘˜π‘›ξ‚βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘₯π‘˜(1βˆ’π‘₯)π‘›βˆ’π‘˜,(1.7) conventionally, the name β€œπ‘ž-Bernstein polynomials” is reserved for the case π‘žβ‰ 1.

Based on the π‘ž-Bernstein polynomials, the π‘ž-Bernstein operator on 𝐢[0,1] is given by 𝐡𝑛,π‘žβˆΆπ‘“β†¦π΅π‘›,π‘ž(𝑓;β‹…).(1.8) A detailed review of the results on the π‘ž-Bernstein polynomials along with an extensive bibliography has been provided in [2]. In this field, new results concerning the properties of the π‘ž-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 π‘ž-Bernstein polynomials is attributed to the fact that they are closely related to the π‘ž-binomial and the π‘ž-deformed Poisson probability distributions (cf. [9]). The π‘ž-binomial distribution plays an important role in the π‘ž-boson theory, providing a π‘ž-deformation for the quantum harmonic formalism. More specifically, it has been used to construct the binomial state for the π‘ž-boson. Meanwhile, the π‘ž-deformed Poisson distribution, which is the limit form of π‘ž-binomial one, defines the energy distribution in a π‘ž-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 π‘ž-Bernstein polynomials retain some of the properties of the classical Bernstein polynomials. For example, they possess the end-point interpolation property: 𝐡𝑛,π‘ž(𝑓;0)=𝑓(0),𝐡𝑛,π‘ž(𝑓;1)=𝑓(1),𝑛=1,2,…,π‘ž>0,(1.9) and leave the linear functions invariant: 𝐡𝑛,π‘ž(π‘Žπ‘‘+𝑏;π‘₯)=π‘Žπ‘₯+𝑏,𝑛=1,2,…,π‘ž>0.(1.10) In addition, the π‘ž-Bernstein basic polynomials (1.2) satisfy the identity π‘›ξ“π‘˜=0π‘π‘›π‘˜(π‘ž;π‘₯)=1βˆ€π‘›=1,2,…,βˆ€π‘ž>0.(1.11) Furthermore, the π‘ž-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 π‘ž-Bernstein polynomials for π‘žβ‰ 1 are essentially different from those of the classical ones. What is more, the cases 0<π‘ž<1 and π‘ž>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<π‘ž<1,  𝐡𝑛,π‘ž are positive linear operators on 𝐢[0,1], whereas for π‘ž>1, no positivity occurs. In addition, the case π‘ž>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 ‖𝐡𝑛,π‘žβ€– increases rather rapidly in both 𝑛 and π‘ž. Namely, ‖‖𝐡𝑛,π‘žβ€–β€–βˆΌ2π‘’β‹…π‘žπ‘›(π‘›βˆ’1)/2𝑛asπ‘›β†’βˆž,π‘žβ†’+∞.(1.12) This puts serious obstacles in the analysis of the convergence for π‘ž>1. The challenge has inspired some papers by a number of authors dealing with the convergence of π‘ž-Bernstein polynomials in the case π‘ž>1 (see, e.g., [7, 17]). However, there are still many open problems related to the behavior of the π‘ž-Bernstein polynomials with π‘ž>1 (see the list of open problems in [2]).

In this paper, it is shown that the time scale π•π‘ž=ξ€½π‘žβˆ’π‘—ξ€Ύβˆžπ‘—=0βˆͺ{0}(1.13) is the β€œminimal” set of convergence for the π‘ž-Bernstein polynomials of continuous functions with π‘ž>1, in the sense that every sequence {𝐡𝑛,π‘ž(𝑓;π‘₯)} converges uniformly on π•π‘ž. Moreover, it is proved that π•π‘ž 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 π‘ž-Bernstein polynomials on the time scale π•π‘ž. Section 3 is devoted to the π‘ž-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 π‘ž-Bernstein Polynomials on π•π‘ž

In this paper, π‘ž>1 is considered fixed. It has been shown in [15], that, if a function 𝑓 is analytic in π·πœ€={π‘§βˆΆ|𝑧|<1+πœ€}, then it is uniformly approximated by its π‘ž-Bernstein polynomials on any compact set in π·πœ€, and, in particular, on [0,1].

In this study, attention is focused on the π‘ž-Bernstein polynomials of β€œbad” 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 π‘ž-Bernstein polynomials on [0,1]. Moreover, their π‘ž-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 {𝐡𝑛,π‘ž(𝑓;π‘₯)} may occur everywhere outside of π•π‘ž, which is a β€œminimal” set of convergence.

However, in spite of this negative information, it will be shown that, for any π‘“βˆˆπΆ[0,1], the sequence of its π‘ž-Bernstein polynomials converges uniformly on the time scale π•π‘ž.

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

Theorem 2.1. Let π‘“βˆˆπΆ[0,1]. Then ||𝐡𝑛,π‘žξ€·π‘“;π‘žβˆ’π‘—ξ€Έξ€·π‘žβˆ’π‘“βˆ’π‘—ξ€Έ||≀2πœ”π‘“βŽ›βŽœβŽœβŽξƒŽπ‘žβˆ’π‘—ξ€·1βˆ’π‘žβˆ’π‘—ξ€Έ[𝑛]π‘žβŽžβŽŸβŽŸβŽ ,π‘—βˆˆβ„€+,(2.1) where πœ”π‘“ is the modulus of continuity of 𝑓 on [0,1].

Corollary 2.2. If π‘—βˆˆβ„€+, then ||𝐡𝑛,π‘žξ€·π‘“;π‘žβˆ’π‘—ξ€Έξ€·π‘žβˆ’π‘“βˆ’π‘—ξ€Έ||≀2πœ”π‘“ξƒ©12√[𝑛]π‘žξƒͺ,(2.2) that is, 𝐡𝑛,π‘ž(𝑓;π‘₯) converges uniformly to 𝑓(π‘₯) on the time scale π•π‘ž.

Proof. The proof is rather straightforward. First, notice that π‘π‘›π‘˜(π‘ž;π‘žβˆ’π‘—)β‰₯0 for all 𝑛,π‘˜,𝑗, while βˆ‘π‘›π‘˜=0π‘π‘›π‘˜(π‘ž;π‘žβˆ’π‘—)=1 by virtue of (1.11). Then ||𝐡𝑛,π‘žξ€·π‘“;π‘žβˆ’π‘—ξ€Έξ€·π‘žβˆ’π‘“βˆ’π‘—ξ€Έ||β‰€π‘›ξ“π‘˜=0||||𝑓[π‘˜]π‘ž[𝑛]π‘žξ‚Άξ€·π‘žβˆ’π‘“βˆ’π‘—ξ€Έ||||π‘π‘›π‘˜ξ€·π‘ž;π‘žβˆ’π‘—ξ€Έβ‰€π‘›ξ“π‘˜=0πœ”π‘“ξƒ©||||[π‘˜]π‘ž[𝑛]π‘žβˆ’π‘žβˆ’π‘—||||ξƒͺπ‘π‘›π‘˜ξ€·π‘ž;π‘žβˆ’π‘—ξ€Έβ‰€πœ”π‘“(𝛿)π‘›ξ“π‘˜=0ξƒ―11+𝛿2ξ‚΅[π‘˜]π‘ž[𝑛]π‘žβˆ’π‘žβˆ’π‘—ξ‚Ά2ξƒ°π‘π‘›π‘˜ξ€·π‘ž;π‘žβˆ’π‘—ξ€Έ(2.3) for any 𝛿>0. Plain calculations (see, e.g., [13], formula (2.7)) show that 𝐡𝑛,π‘žξ€·(π‘‘βˆ’π‘₯)2ξ€Έ=;π‘₯π‘₯(1βˆ’π‘₯)[𝑛]π‘ž,(2.4) which implies that ||𝐡𝑛,π‘žξ€·π‘“;π‘žβˆ’π‘—ξ€Έξ€·π‘žβˆ’π‘“βˆ’π‘—ξ€Έ||β‰€πœ”π‘“ξƒ―1(𝛿)β‹…1+𝛿2β‹…π‘žβˆ’π‘—ξ€·1βˆ’π‘žβˆ’π‘—ξ€Έ[𝑛]π‘žξƒ°.(2.5) Then, one can immediately derive the result by choosing βˆšπ›Ώ=π‘žβˆ’π‘—(1βˆ’π‘žβˆ’π‘—)/[𝑛]π‘ž.

Remark 2.3. In [7], Wu has shown that if π‘“βˆˆπΆ1[0,1], then for any π‘—βˆˆβ„€+, one has: ||𝐡𝑛,π‘žξ€·π‘“;π‘žβˆ’π‘—ξ€Έξ€·π‘žβˆ’π‘“βˆ’π‘—ξ€Έ||≀𝐢𝑗(π‘žβˆ’π‘›),π‘›β†’βˆž,whereπΆπ‘—β†’βˆžasπ‘—β†’βˆž.(2.6) The condition π‘“βˆˆπΆ1[0,1] cannot be left out completely, as the following example shows.

Example 2.4. Consider a function π‘“βˆˆπΆ[0,1] satisfying ⎧βŽͺ⎨βŽͺ⎩0𝑓(π‘₯)=ifξ€Ίπ‘₯∈0,π‘žβˆ’2ξ€»βˆͺξ€·π‘žβˆ’1ξ€»,ξ€·π‘ž,1βˆ’1ξ€Έβˆ’π‘₯𝛼ifξƒ¬ξ€·π‘žπ‘₯βˆˆβˆ’2+π‘žβˆ’1ξ€Έ2,π‘žβˆ’1ξƒ­,(2.7) where 0<𝛼<1. Then, for 𝑛 large enough, we have 𝐡𝑛,π‘žξ€·π‘“;π‘žβˆ’1ξ€Έξ€·π‘žβˆ’π‘“βˆ’1ξ€Έξ‚΅[]=π‘“π‘›βˆ’1π‘ž[𝑛]π‘žξ‚Άπ‘π‘›,π‘›βˆ’1ξ€·π‘ž;π‘žβˆ’1ξ€Έ=ξ‚΅π‘žβˆ’1π‘žξ‚Άπ›Όβ‹…(π‘žπ‘›βˆ’1)1βˆ’π›Όπ‘žπ‘›β‰₯πΆπ‘žβˆ’π‘›π›Ό,(2.8) where 𝐢 is a positive constant independent from 𝑛.

As it has been already mentioned, the behavior of the π‘ž-Bernstein polynomials in the case π‘ž>1 outside of the time scale π•π‘ž may be rather unpredictable. The next theorem shows that the sequence {𝐡𝑛,π‘ž(𝑓;π‘₯)} may be divergent for all π‘₯βˆˆβ„β§΅π•π‘ž.

Theorem 2.5. Let 𝑓(π‘₯)=π‘₯𝛼,  0<𝛼≀1/2. If π‘žβ‰₯2, then 𝐡𝑛,π‘ž(𝑓;π‘₯)β†’βˆžasπ‘›β†’βˆžβˆ€π‘₯βˆˆβ„β§΅π•π‘ž.(2.9)

Proof. The π‘ž-Bernstein polynomial of 𝑓 is 𝐡𝑛,π‘ž(𝑓;π‘₯)=π‘›ξ“π‘˜=0ξ‚΅[π‘˜]π‘ž[𝑛]π‘žξ‚Άπ›Όπ‘π‘›π‘˜1(π‘ž;π‘₯)=[𝑛]π›Όπ‘žπ‘›ξ“π‘˜=1[π‘˜]π›Όπ‘žπ‘π‘›π‘˜(π‘ž;π‘₯).(2.10) Since for π‘˜=1,2,…,π‘›βˆ’1 one has π‘π‘›π‘˜(π‘ž;π‘₯)=(π‘žπ‘›ξ€·π‘žβˆ’1)β‹―π‘›βˆ’π‘˜+1ξ€Έβˆ’1ξ€·π‘žπ‘˜ξ€Έπ‘₯βˆ’1β‹―(π‘žβˆ’1)π‘˜(π‘₯;π‘ž)π‘›βˆ’π‘˜=π‘ž(2π‘›βˆ’π‘˜+1)π‘˜/2(π‘žβˆ’π‘›;π‘ž)π‘˜ξ€·π‘žπ‘˜ξ€Έβˆ’1β‹―(π‘žβˆ’1)(βˆ’1)π‘›βˆ’π‘˜π‘ž(π‘›βˆ’π‘˜)(π‘›βˆ’π‘˜βˆ’1)/2π‘₯𝑛1π‘₯;1π‘žξ‚Άπ‘›βˆ’π‘˜=(βˆ’1)π‘›π‘žπ‘›(π‘›βˆ’1)/2π‘₯𝑛⋅(βˆ’1)π‘˜π‘žπ‘˜(π‘žβˆ’π‘›;π‘ž)π‘˜ξ€·π‘žπ‘˜ξ€Έβ‹…ξ‚΅1βˆ’1β‹―(π‘žβˆ’1)π‘₯;1π‘žξ‚Άπ‘›βˆ’π‘˜,(2.11) it follows that 𝐡𝑛,π‘ž(𝑓;π‘₯)=(βˆ’1)π‘›π‘žπ‘›(π‘›βˆ’1)/2π‘₯𝑛[𝑛]π›Όπ‘žβ‹…π‘‡(𝑛,π‘ž,π‘₯),(2.12) where 𝑇(𝑛;π‘ž;π‘₯)∢=(βˆ’1)𝑛[𝑛]π›Όπ‘žπ‘žπ‘›(π‘›βˆ’1)/2+π‘›βˆ’1ξ“π‘˜=0(βˆ’1)π‘˜[π‘˜]π›Όπ‘žπ‘žπ‘˜(π‘žβˆ’π‘›;π‘ž)π‘˜ξ€·π‘žπ‘˜ξ€Έβ‹…ξ‚΅1βˆ’1β‹―(π‘žβˆ’1)π‘₯;1π‘žξ‚Άπ‘›βˆ’π‘˜.(2.13) Obviously, limπ‘›β†’βˆžπ‘žπ‘›(π‘›βˆ’1)/2π‘₯𝑛[𝑛]π›Όπ‘ž=∞foranyπ‘₯β‰ 0.(2.14) As such, the theorem will be proved if it is shown that limπ‘›β†’βˆžπ‘‡(𝑛,π‘ž,π‘₯)β‰ 0forπ‘₯βˆ‰π•π‘ž.(2.15) As limπ‘›β†’βˆž(π‘žβˆ’π‘›(π‘›βˆ’1)/2(βˆ’1)𝑛[𝑛]π›Όπ‘ž)=0, it suffices to prove that limβˆžπ‘›β†’βˆžξ“π‘˜=0π‘π‘˜π‘›β‰ 0whenπ‘₯βˆ‰π•π‘ž,(2.16) where π‘π‘˜π‘›βŽ§βŽͺ⎨βŽͺ⎩∢=(βˆ’1)π‘˜[π‘˜]π›Όπ‘žπ‘žπ‘˜(π‘žβˆ’π‘›;π‘ž)π‘˜ξ€·π‘žπ‘˜ξ€Έβ‹…ξ‚΅1βˆ’1β‹―(π‘žβˆ’1)π‘₯;1π‘žξ‚Άπ‘›βˆ’π‘˜if0π‘˜β‰€π‘›βˆ’1,ifπ‘˜β‰₯𝑛.(2.17) The fact that (π‘žβˆ’π‘›;π‘ž)π‘˜β‰€1 and the inequality ||||ξ‚΅1π‘₯;1π‘žξ‚Άπ‘›βˆ’π‘˜||||β‰€ξ‚΅βˆ’1;1|π‘₯|π‘žξ‚Άπ‘›βˆ’π‘˜β‰€ξ‚΅βˆ’1;1|π‘₯|π‘žξ‚Άβˆž(2.18) lead to ||π‘π‘˜π‘›||β‰€π‘žπ‘˜[π‘˜]π›Όπ‘žξ€·π‘žπ‘˜ξ€Έξ‚΅βˆ’1βˆ’1β‹―(π‘žβˆ’1);1|π‘₯|π‘žξ‚Άβˆž=βˆΆπ‘‘π‘˜.(2.19) Now, since limπ‘›β†’βˆžπ‘π‘˜π‘›=(βˆ’1)π‘˜π‘žπ‘˜[π‘˜]π›Όπ‘žξ€·π‘žπ‘˜ξ€Έξ‚΅1βˆ’1β‹―(π‘žβˆ’1)π‘₯;1π‘žξ‚Άβˆž,(2.20) and the series βˆ‘βˆžπ‘˜=0π‘‘π‘˜ is convergent, the Lebesgues dominated convergence theorem implies limβˆžπ‘›β†’βˆžξ“π‘˜=0π‘π‘˜π‘›=βˆžξ“π‘˜=0limπ‘›β†’βˆžπ‘π‘˜π‘›=1(π‘žβˆ’1)𝛼1π‘₯;1π‘žξ‚Άβˆžβ‹…βˆžξ“π‘˜=0(βˆ’1)π‘˜π‘Žπ‘˜,(2.21) where π‘Žπ‘˜=π‘žπ‘˜(π‘žπ‘˜βˆ’1)𝛼/(π‘žπ‘˜βˆ’1)β‹―(π‘žβˆ’1),π‘˜=1,2,…. Moreover, 1(π‘žβˆ’1)𝛼1π‘₯;1π‘žξ‚Άβˆžβ‰ 0wheneverπ‘₯βˆ‰π•π‘ž.(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 π‘Žπ‘˜+1<π‘Žπ‘˜,β€‰β€‰π‘˜=1,2,… for π‘žβ‰₯2. Since π‘Žπ‘˜+1π‘Žπ‘˜=π‘žξ€·π‘žπ‘˜+1ξ€Έβˆ’11βˆ’π›Όξ€·π‘žπ‘˜ξ€Έβˆ’1𝛼,(2.23) for π‘˜β‰₯2 it follows that π‘Žπ‘˜+1π‘Žπ‘˜β‰€π‘žξ€·π‘ž3ξ€Έβˆ’11βˆ’π›Όξ€·π‘ž2ξ€Έβˆ’1π›Όβ‰€π‘žξ€·π‘ž(π‘žβˆ’1)2ξ€Έ+π‘ž+11βˆ’π›Ό(π‘ž+1)π›Όβ‰€π‘ž(π‘žβˆ’1)(π‘ž+1)<1.(2.24) Notice that (2.24) holds for any π›Όβˆˆ(0,1). In addition, if π‘˜=1, then π‘Ž2π‘Ž1=π‘žξ€·π‘ž2ξ€Έβˆ’11βˆ’π›Ό(π‘žβˆ’1)𝛼=π‘ž(π‘žβˆ’1)(π‘ž+1)1βˆ’π›Ό.(2.25) The function in the r.h.s. is monotone decreasing in π‘ž, so π‘Ž2π‘Ž1≀21β‹…31βˆ’π›Όβ‰€23√9<1.(2.26) Thus, {π‘Žπ‘˜}βˆžπ‘˜=1 is a strictly decreasing sequence. Since all (π‘Ž2π‘˜βˆ’1βˆ’π‘Ž2π‘˜) are strictly positive, it follows that βˆžξ“π‘˜=0(βˆ’1)π‘˜π‘Žπ‘˜π‘Ž=βˆ’ξ€Ίξ€·1βˆ’π‘Ž2ξ€Έ+ξ€·π‘Ž3βˆ’π‘Ž4ξ€Έξ€·π‘Ž+β‹―+2π‘˜βˆ’1βˆ’π‘Ž2π‘˜ξ€Έξ€»+β‹―<0.(2.27)
Case 2. 1/3≀𝛼≀1/2.
Estimate (2.24) implies that βˆ‘βˆžπ‘˜=5(βˆ’1)π‘˜π‘Žπ‘˜<0. To prove the theorem, it suffices to show that π‘Ž1βˆ’π‘Ž2+π‘Ž3βˆ’π‘Ž4>0 when π‘žβ‰₯2. Denoting π‘Žπ‘–=(π‘ž(π‘žβˆ’1)𝛼/(π‘žβˆ’1))𝑔𝑖(π‘ž), 𝑖=1,2,3,4, we write the following: π‘Ž1βˆ’π‘Ž2+π‘Ž3βˆ’π‘Ž4=π‘ž(π‘žβˆ’1)π›Όξ€Ίπ‘”π‘žβˆ’11(π‘ž)βˆ’π‘”2(π‘ž)+𝑔3(π‘ž)βˆ’π‘”4ξ€»π‘ž(π‘ž)=∢(π‘žβˆ’1)π›Όπ‘žβˆ’1𝐾(π‘ž).(2.28) We are left to show that 𝐾(π‘ž) is strictly positive for the specified values of π‘ž and 𝛼. First of all, notice that 𝑔1(π‘ž)=1, while 𝑔2(π‘ž),𝑔3(π‘ž), and 𝑔4(π‘ž) are strictly decreasing in π‘ž on (0,+∞). Hence, for π‘žβˆˆ[2,5/2], 𝐾(π‘ž)β‰₯1βˆ’π‘”1(2)+𝑔2ξ‚€52ξ‚βˆ’π‘”32(2)=1βˆ’3β‹…3𝛼+200ξ‚€2457394ξ‚π›Όβˆ’8315β‹…15𝛼=∢𝐿(𝛼).(2.29) The function 𝐿(𝛼) is strictly decreasing on [1/3,1/2]. Indeed, πΏξ…ž2(𝛼)=βˆ’3β‹…3𝛼ln3+200ξ‚€2457394𝛼ln394ξ‚βˆ’8315β‹…15𝛼ln15(2.30) and, for π›Όβˆˆ[1/3,1/2], πΏξ…ž2(𝛼)β‰€βˆ’3β‹…31/3ln3+200ξ‚€24573941/2ξ‚€ln394ξ‚βˆ’8315β‹…151/3ln15β‰€βˆ’0.4332<0,(2.31) whence 𝐿(𝛼)β‰₯𝐿(1/2)β‰₯1.096Γ—10βˆ’3>0 for π›Όβˆˆ[1/3,1/2].
Similarly, for π‘žβˆˆ[5/2,3], 𝐾(π‘ž)β‰₯1βˆ’π‘”1ξ‚€52+𝑔2(3)βˆ’π‘”3ξ‚€52=1βˆ’10β‹…ξ‚€7212𝛼+9208β‹…13π›Όβˆ’8000β‹…ξ‚€14963132038𝛼=βˆΆπ‘€(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)β‰₯0.238>0, it follows that 𝑀(𝛼)>0 for all π›Όβˆˆ[1/3,1/2].
Finally, for π‘žβˆˆ[3,+∞), we obtain 𝐾(π‘ž)β‰₯1βˆ’π‘”1(3)βˆ’π‘”33(3)=1βˆ’8β‹…4π›Όβˆ’2716640β‹…40𝛼=βˆΆπ‘(𝛼).(2.33) Obviously, 𝑁(𝛼) is a strictly decreasing function for all π›Όβˆˆβ„, whence, for π›Όβˆˆ[1/3,1/2], ξ‚€1𝑁(𝛼)β‰₯𝑁2β‰₯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 π›Όβˆˆ(0,1) and π‘žβ‰₯π‘ž0(𝛼).

An illustrative example is supplied below.

Example 2.7. Let 𝑓(π‘₯)=3√π‘₯. The graphs of 𝑦=𝑓(π‘₯) and 𝑦=𝐡𝑛,π‘ž(𝑓;π‘₯) for π‘ž=2 and 𝑛=4,5 are exhibited in Figure 1. Similarly, Figure 2 represents the graphs of 𝑦=𝑓(π‘₯) and 𝑦=𝐡𝑛,π‘ž(𝑓;π‘₯) for π‘ž=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,π‘ž,π‘₯)∢=𝐡𝑛,π‘ž(𝑓;π‘₯)βˆ’π‘“(π‘₯) with π‘ž=2 at some points π‘₯∈[0,1]. The points are taken both in π•π‘ž and in [0,1]β§΅π•π‘ž. It can be observed from Table 1 that, while at the points π‘₯βˆˆπ•π‘ž, the values of the error function are close to 0, at the points π‘₯βˆ‰π•π‘ž, the values of the error function may be very large in magnitude.

Table 1: The values of 𝐸(𝑛,π‘ž,π‘₯)=𝐡𝑛,π‘ž(𝑓;π‘₯)βˆ’π‘“(π‘₯) at some points π‘₯∈[0,1].
Figure 1: Graphs of 𝑦=𝑓(π‘₯) and 𝑦=𝐡𝑛,2(𝑓;π‘₯),  𝑛=4,5.
Figure 2: Graphs of 𝑦=𝑓(π‘₯) and 𝑦=𝐡𝑛,2(𝑓;π‘₯),  𝑛=6,7.

Remark 2.8. Table 1 also shows that while the error function changes its sign for different values of π‘₯, for π‘₯=π‘žβˆ’π‘—βˆˆπ•π‘ž, its values are negative, that is, 𝐡𝑛,π‘ž(𝑑1/3;π‘žβˆ’π‘—)<𝑓(π‘žβˆ’π‘—) for π‘žβˆ’π‘—βˆˆπ•π‘ž. This is a particular case of the following statement.

Theorem 2.9. Let π‘ž>1. If 𝑓(π‘₯) is convex (concave) on [0,1], then 𝐡𝑛,π‘žξ€·π‘“;π‘žβˆ’π‘—ξ€Έξ€·π‘žβ‰₯π‘“βˆ’π‘—ξ€Έξ€·correspondingly𝐡𝑛,π‘žξ€·π‘“;π‘žβˆ’π‘—ξ€Έξ€·π‘žβ‰€π‘“βˆ’π‘—,ξ€Έξ€Έ(2.35) for all π‘žβˆ’π‘—βˆˆπ•π‘ž.

Proof. It can be readily seen from (1.10) and (1.11) that π‘›ξ“π‘˜=0π‘π‘›π‘˜ξ€·π‘ž;π‘žβˆ’π‘—ξ€Έ=1,π‘›ξ“π‘˜=0[π‘˜]π‘ž[𝑛]π‘žπ‘π‘›π‘˜ξ€·π‘ž;π‘žβˆ’π‘—ξ€Έ=π‘žβˆ’π‘—,(2.36) while π‘π‘›π‘˜(π‘ž;π‘žβˆ’π‘—)β‰₯0. By virtue of Jensen's inequality, if 𝑓 is convex on [0,1], then whenever π‘›βˆˆβ„• and π‘₯0,π‘₯1,…,π‘₯π‘›βˆˆ[π‘Ž,𝑏], there holds the following: π‘›ξ“π‘˜=0πœ†π‘˜π‘“ξ€·π‘₯π‘˜ξ€Έξƒ©β‰₯π‘“π‘›ξ“π‘˜=0πœ†π‘˜π‘₯π‘˜ξƒͺ.(2.37) for all πœ†0,πœ†1,…,πœ†π‘›β‰₯0 satisfying βˆ‘π‘›π‘˜=0πœ†π‘˜=1. Setting π‘₯π‘˜=[π‘˜]π‘ž[𝑛]π‘ž,πœ†π‘˜=π‘π‘›π‘˜ξ€·π‘ž;π‘žβˆ’π‘—ξ€Έ,π‘˜=0,1,…,𝑛,(2.38) and observing that π‘›ξ“π‘˜=0πœ†π‘˜π‘“ξ€·π‘₯π‘˜ξ€Έ=𝐡𝑛,π‘žξ€·π‘“;π‘žβˆ’π‘—ξ€Έ,(2.39) the required result is derived.

Example 2.10. Let π‘“βŽ§βŽͺ⎨βŽͺβŽ©π‘ž(π‘₯)=2π‘₯if0≀π‘₯β‰€π‘žβˆ’2,π‘ž2ξ€·π‘žπ‘₯βˆ’2ξ€Έπ‘₯βˆ’12π‘ž2βˆ’1ifπ‘žβˆ’2<π‘₯≀1.(2.40) The function is concave on [0,1] and, hence, according to the previous results, 𝐡𝑛,π‘ž(𝑓;π‘žβˆ’π‘—)→𝑓(π‘žβˆ’π‘—) as π‘›β†’βˆž from below for all π‘—βˆˆβ„€+. To examine the behavior of polynomials 𝐡𝑛,π‘ž(𝑓;π‘₯) for π‘₯βˆ‰π•π‘ž, consider the auxiliary function: 𝑔(π‘₯)=𝑓(π‘₯)βˆ’π‘ž2⎧βŽͺ⎨βŽͺ⎩0π‘₯=if0≀π‘₯β‰€π‘žβˆ’2,βˆ’ξ€·π‘ž2ξ€Έπ‘₯βˆ’12π‘ž2βˆ’1ifπ‘žβˆ’2<π‘₯≀1.(2.41) Since [π‘›βˆ’π‘˜]π‘ž/[𝑛]π‘žβ‰€π‘žβˆ’π‘˜ for π‘˜=0,1,…,𝑛, and [π‘›βˆ’1]π‘ž/[𝑛]π‘žβ‰₯π‘žβˆ’2 whenever π‘žπ‘›β‰₯π‘ž+1, it follows that, for sufficiently large 𝑛, 𝐡𝑛,π‘žξ‚΅[](𝑔;π‘₯)=π‘”π‘›βˆ’1π‘ž[𝑛]π‘žξ‚Άπ‘π‘›,π‘›βˆ’1(π‘ž;π‘₯)+𝑔(1)𝑝𝑛𝑛(π‘₯).(2.42) Plain computations reveal 𝑔[]π‘›βˆ’1π‘ž[𝑛]π‘žξ‚Ά=βˆ’(π‘žβˆ’1)(π‘žπ‘›)βˆ’π‘žβˆ’12(π‘žπ‘›βˆ’1)2,(π‘ž+1)(2.43) yielding 𝐡𝑛,π‘ž(𝑔;π‘₯)=βˆ’(π‘žπ‘›)βˆ’π‘žβˆ’12(π‘žπ‘›π‘₯βˆ’1)(π‘ž+1)π‘›βˆ’1ξ€·π‘ž(1βˆ’π‘₯)βˆ’2ξ€Έπ‘₯βˆ’1𝑛.(2.44) Consequently, for π‘₯βˆ‰π•π‘ž, one obtains limπ‘›β†’βˆžπ΅π‘›,π‘žβŽ§βŽͺ⎨βŽͺ⎩0(𝑔;π‘₯)=if|π‘₯|<π‘žβˆ’1,∞if|π‘₯|>π‘žβˆ’1.(2.45) Since, by (1.10), 𝐡𝑛,π‘ž(𝑓;π‘₯)=π‘ž2π‘₯+𝐡𝑛,π‘ž(𝑔;π‘₯), it follows that: limπ‘›β†’βˆžπ΅π‘›,π‘žβŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π‘ž(𝑓;π‘₯)=2π‘₯if|π‘₯|<π‘žβˆ’1,∞if|π‘₯|>π‘žβˆ’1π‘ž,π‘₯β‰ 1,2+1π‘ž+1ifπ‘₯=π‘žβˆ’1,1ifπ‘₯=1.(2.46) For π‘₯=βˆ’π‘žβˆ’1, the limit does not exist. Additionally, it is not difficult to see that 𝐡𝑛,π‘ž(𝑓;π‘₯)→𝑓(π‘₯) as π‘›β†’βˆž uniformly on any compact set inside (βˆ’1/π‘ž2,1/π‘ž2), while on any interval outside of (βˆ’1/π‘ž2,1/π‘ž2), the function 𝑓(π‘₯) is not approximated by its π‘ž-Bernstein polynomials. This agrees with the result from [17], Theorem 2.3. The graphs of 𝑓(π‘₯) and 𝐡𝑛,π‘ž(𝑓;π‘₯) for π‘ž=2,  𝑛=5 and 8 on [0,1] are given in Figure 3. The values of the error function at some points π‘₯βˆˆπ•π‘ž and at some exemplary points π‘₯βˆ‰π•π‘ž are given in Table 2.

Table 2: The values of 𝐸(𝑛,π‘ž,π‘₯)=𝐡𝑛,π‘ž(𝑓;π‘₯)βˆ’π‘“(π‘₯) at some points π‘₯∈[0,1].
Figure 3: Graphs of 𝑦=𝑓(π‘₯) and𝑦=𝐡𝑛,2(𝑓;π‘₯), 𝑛=5,8.

Remark 2.11. Following Charalambides [9], consider a sequence of random variables {𝑋𝑛(𝑗)}βˆžπ‘›=1 possessing the distributions 𝑃𝑛(𝑗) given by 𝐏𝑋𝑛(𝑗)=[]π‘›βˆ’π‘˜π‘ž[𝑛]π‘žξ‚Ό=𝑝𝑛,π‘›βˆ’π‘˜ξ€·π‘ž;π‘žβˆ’π‘—ξ€Έ,π‘˜=0,1,…,𝑛.(2.47) Let 𝐼(π‘žβˆ’π‘—) denote a random variable with the 𝛿-distribution concentrated at π‘žβˆ’π‘—. Theorem 2.1 implies that 𝑋𝑛(𝑗)→𝐼(π‘žβˆ’π‘—) in distribution.

Generally speaking, Theorem 2.1 shows that the π‘ž-Bernstein polynomials with π‘ž>1 possess an β€œinterpolation-type” property on π•π‘ž. Information on interpolation of functions with nodes on a geometric progression can be found in, for example, [18] by Schoenberg.

3. On the π‘ž-Bernstein Polynomials of the Weierstrass-Type Functions

In this section, the π‘ž-Bernstein polynomials of the functions with β€œbad” smoothness are considered. Let πœ‘(π‘₯)∈𝐢[βˆ’1,1] satisfy the condition: πœ‘(0)>πœ‘(π‘₯)for[]π‘₯βˆˆβˆ’1,1⧡{0}.(3.1) The letter πœ‘ will also denote a 2-periodic continuation of πœ‘(π‘₯) on (βˆ’βˆž,∞).

Definition 3.1. Let π‘Ž,π‘βˆˆβ„ satisfy 0<π‘Ž<1<π‘Žπ‘. A function 𝑓(π‘₯) is said to be Weierstrass-type if 𝑓(π‘₯)=βˆžξ“π‘˜=0π‘Žπ‘˜πœ‘ξ€·π‘π‘˜π‘₯ξ€Έ.(3.2) Notice that 𝑓(π‘₯) is continuous if and only if πœ‘(βˆ’1)=πœ‘(1). For πœ‘(π‘₯)=cosπœ‹π‘₯ and a special choice of π‘Ž 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 πœ‘(π‘₯)=1βˆ’|π‘₯|, 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(π‘₯) is a Weierstrass-type function, then the sequence 𝐡𝑛,π‘ž(𝑓;π‘₯) of its π‘ž-Bernstein polynomials is not uniformly bounded on any interval [0,𝑐].

Proof. To prove the theorem, the following representation of π‘ž-Bernstein polynomials (see [15], formulae (6) and (7)) is used: 𝐡𝑛,π‘ž(𝑓;π‘₯)=π‘›ξ“π‘˜=0πœ†π‘˜π‘›π‘“ξ‚Έ10;[𝑛]π‘ž[π‘˜];…;π‘ž[𝑛]π‘žξ‚Ήπ‘₯π‘˜,(3.3) where πœ†0𝑛=πœ†1𝑛=1,πœ†π‘˜π‘›=ξ‚΅11βˆ’[𝑛]π‘žξ‚Άβ‹―ξ‚΅[]1βˆ’π‘˜βˆ’1π‘ž[𝑛]π‘žξ‚Ά,π‘˜=2,…,𝑛,(3.4) and 𝑓[π‘₯0;π‘₯1;…;π‘₯π‘˜] denote the divided differences of 𝑓, that is, 𝑓π‘₯0ξ€»ξ€·π‘₯=𝑓0ξ€Έξ€Ίπ‘₯,𝑓0;π‘₯1ξ€»=𝑓π‘₯1ξ€Έξ€·π‘₯βˆ’π‘“0ξ€Έπ‘₯1βˆ’π‘₯0𝑓π‘₯,…,0;π‘₯1;…;π‘₯π‘˜ξ€»=𝑓π‘₯1;…;π‘₯π‘˜ξ€»ξ€Ίπ‘₯βˆ’π‘“0;…;π‘₯π‘˜βˆ’1ξ€»π‘₯π‘˜βˆ’π‘₯0.(3.5) When π‘ž=1, the well-known representation for the classical Bernstein polynomials is recovered and the numbers πœ†π‘˜π‘› 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 π‘žβ‰ 1 in [15].
Clearly, it suffices to consider the case 0<𝑐<1. From (3.3), it follows that π΅ξ…žπ‘›,π‘ž(𝑓;0)=πœ†1𝑛𝑓10;[𝑛]π‘žξ‚Ή=[𝑛]π‘žξ‚»π‘“ξ‚΅1[𝑛]π‘žξ‚Άξ‚Ό,βˆ’π‘“(0)(3.6) and, hence, ||π΅ξ…žπ‘›,π‘ž(||=[𝑛]𝑓;0)π‘žξ‚»ξ‚΅1𝑓(0)βˆ’π‘“[𝑛]π‘ž=[𝑛]ξ‚Άξ‚Όπ‘žβˆžξ“π‘˜=0π‘Žπ‘˜ξ‚»ξ‚΅π‘πœ‘(0)βˆ’πœ‘π‘˜[𝑛]π‘ž.ξ‚Άξ‚Ό(3.7) What remains is to find a lower bound for |π΅ξ…žπ‘›,π‘ž(𝑓;0)|. Due to (3.1), all terms of the series are nonnegative and, therefore, ||π΅ξ…žπ‘›,π‘ž||β‰₯[𝑛](𝑓;0)π‘žπ‘Žπ‘—ξ‚»ξ‚΅π‘πœ‘(0)βˆ’πœ‘π‘—[𝑛]π‘žξ‚Άξ‚Όforany𝑗=0,1,…(3.8) Let 𝑗=𝑗𝑛 be chosen in such a way that 1𝑏<𝑏𝑗𝑛[𝑛]π‘žβ‰€1.(3.9) For 𝑛>𝑏, such a choice is possible because, in this case, inequality (3.9) implies that [𝑛]0<lnπ‘žlnπ‘βˆ’1<𝑗𝑛≀[𝑛]lnπ‘ž.ln𝑏(3.10) Since the length of the interval (ln[𝑛]π‘ž/lnπ‘βˆ’1,ln[𝑛]π‘ž/ln𝑏] is 1, there is a positive integer, say, 𝑗𝑛, such that π‘—π‘›βˆˆ(ln[𝑛]π‘ž/lnπ‘βˆ’1,ln[𝑛]π‘ž/ln𝑏]. The obvious inequality [𝑛]π‘ž>π‘žπ‘›βˆ’1 implies the following: 𝑗𝑛β‰₯(π‘›βˆ’1)lnπ‘žlnπ‘βˆ’1=βˆΆπ‘›lnπ‘žlnπ‘βˆ’π΄,(3.11) with 𝐴=(lnπ‘ž/ln𝑏)+1 being a positive constant. Then, for 𝑛>𝑏, it follows that ||π΅ξ…žπ‘›,π‘ž||β‰₯[𝑛](𝑓;0)π‘žπ‘Žπ‘—π‘›min[]π‘‘βˆˆ1/𝑏,1[𝑛]{πœ‘(0)βˆ’πœ‘(𝑑)}∢=πœπ‘žπ‘Žπ‘—π‘›,(3.12) where 𝜏>0 due to (3.1). Consequently, ||π΅ξ…žπ‘›,π‘ž(||[𝑛]𝑓;0)β‰₯πœπ‘žπ‘π‘—π‘›(π‘Žπ‘)𝑗𝑛β‰₯𝜏(π‘Žπ‘)𝑗𝑛β‰₯𝜏(π‘Žπ‘)𝑛(lnq/ln𝑏)βˆ’π΄,(3.13) which leads to ||π΅ξ…žπ‘›,π‘ž(||𝑓;0)β‰₯πΆπœŒπ‘›,(3.14) where 𝐢=𝜏(π‘Žπ‘)βˆ’π΄ is a positive constant and 𝜌=(π‘Žπ‘)(lnq/ln𝑏)>1. Now, assume that {𝐡𝑛,π‘ž(𝑓;π‘₯)} is uniformly bounded on [0,𝑐], that is, |𝐡𝑛,π‘ž(𝑓;π‘₯)|≀𝑀 for all π‘₯∈[0,𝑐]. By Markov's Inequality (cf., e.g., [22], Chapter 4, Section 1, pp. 97-98) it follows that ||π΅ξ…žπ‘›,π‘ž||≀(𝑓;0)2𝑀𝑐𝑛2βˆ€π‘›=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 β„Žπ‘, that is: β„Žπ‘(π‘₯)=π‘ξ“π‘˜=0π‘Žπ‘˜πœ‘ξ€·π‘π‘˜π‘₯ξ€Έ.(3.16) Clearly, the function β„Žπ‘ is an approximation of (3.2) satisfying the error estimate 𝐸𝑁||(π‘₯)=𝑓(π‘₯)βˆ’β„Žπ‘||(π‘₯)≀max[]π‘‘βˆˆβˆ’1,1||||π‘Žπœ‘(𝑑)𝑁+1[].1βˆ’π‘Ž,βˆ€π‘₯∈0,1(3.17)

Example 3.3. Let πœ‘(π‘₯)=(cosπœ‹π‘₯),π‘Ž=1/2, and 𝑏=4. For 𝑁=20, one has 𝐸20(π‘₯)≀10βˆ’6. The graphs of β„Ž20(π‘₯) and the associated π‘ž-Bernstein polynomials 𝐡𝑛,π‘ž(β„Ž20;π‘₯) for π‘ž=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=β„Ž20(π‘₯) and 𝑦=𝐡𝑛,2(𝑓;π‘₯),  𝑛=4,5,6.
Figure 5: Graphs of 𝑦=β„Ž20(π‘₯) and 𝑦=𝐡𝑛,2(𝑓;π‘₯),  𝑛=4,5,6.


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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