A basic problem of \((p,q)\) Bernstein type operators

R E S E A R C H

Open Access

A basic problem of

(

p

,

q

)

-Bernstein-type

operators

Qing-Bo Cai

_{and Xiao-Wei Xu}

*_{Correspondence:}

[email protected]

2_{School of Mathematical Sciences,}

Xiamen University, Xiamen, China Full list of author information is available at the end of the article

Abstract

In this note, we give an elaboration of a basic problem on convergence theorem of (p,q)-analogue of Bernstein-type operators. By some classical analysis techniques, we derive an exact class of (pn,qn)-integer satisfying limn→∞[n]pn,qn=∞with

limn→∞pn= 1 and limn→∞qn= 1 under 0 <qn<pn≤1. Our results provide an

erratum to corresponding results on (p,q)-analogue of Bernstein-type operators that appeared in recent literature.

MSC: 41A50

Keywords: (p,q)-integer; Bernstein-type approximation; convergence theorem; equivalent condition

1 Introduction

During the last decades, the applications ofq-calculus emerged as a new area in the field of approximation theory. The rapid development ofq-calculus has led to the discovery of various generalizations of Bernstein polynomials involvingq-integers. A detailed review of the results onq-Bernstein polynomials along with an extensive bibliography is given in []. Theq-Bernstein polynomials are shown to be closely related to theq-deformed binomial distribution []. It plays an important role in theq-boson theory giving aq-deformation of the quantum harmonic formalism []. Theq-analogue of the boson operator calculus has proved to be a powerful tool in theoretical physics. It provides explicit expressions for the representations of the quantum groupSUq() []. Meanwhile, the (p,q)-integers were

introduced in order to generalize or unify several forms ofq-oscillator algebras well known in the earlier physics literature related to the representation theory of single parameter quantum algebras [].

Recently, (p,q)-integers have been introduced into classical linear positive operators to construct new approximation processes. A sequence of (p,q)-analogue of Bernstein operators was first introduced by Mursaleen [, ]. Besides, (p,q)-analogues of Szász-Mirakyan [], Baskakov Kantorovich [], Bleimann-Butzer-Hahn [] and Kantorovich-type Bernstein-Stancu-Schurer [] operators were also considered, see [–]. For fur-ther developments, one can also refer to [, –]. These operators are double parameters corresponding topandqversus single parameterq-Bernstein-type operators [, , ]. The aim of these generalizations is to provide appropriate and powerful tools to applica-tion areas such as numerical analysis, computer-aided geometric design and soluapplica-tions of differential equations (see, e.g., []).

For example, consider the (p,q)-analogue of the Bernstein operators proposed in []. Givenf ∈C[, ] and  <q<p≤, operatorsBn,p,qare defined as follows:

Bn,p,q(f;x)

:=  pn(n–)

n

k=

n k

p,q

pk(k–)_xk

n–k–

s=

ps–qsxf

[k]p,q

pk–n_[n] p,q

, n= , , . . . , (.)

where, for any nonnegative integerkand  <q<p≤, the (p,q)-integer [k]p,qis defined

by

[k]p,q:=pk–+pk–q+· · ·+qk–=

pk_–qk

p–q (k= , , , . . .), []p,q:= , and the (p,q)-factorial [k]p,q! is defined by

[k]p,q! := []p,q[]p,q· · ·[k]p,q (k= , , . . .), []p,q! := .

For integersk,nwith ≤k≤n, the (p,q)-binomial coefficient is defined by

n k

p,q

:= [n]p,q! [k]p,q![n–k]p,q!

.

In general, we expectBn,p,q(f;x) to converge tof(x) asn→ ∞. But we see that, for fixed

value ofpandqwithq∈(, ) andp∈(q, ],

[n]p,q→ or /( –q) asn→ ∞.

To obtain a sequence of generalized (p,q)-analogue Bernstein polynomials which con-verge, we letqn∈(, ) andpn∈(qn, ] depend onn. We then choose a sequence (pn,qn)

such that [n]pn,qn→ ∞asn→ ∞, to ensure thatBn,p,q(f;x) converge tof(x).

The convergence theorems for (p,q)-analogue Bernstein-type operators were estab-lished in some recent papers (see [], Theorem . (Remark .), [], Theorem , and fur-ther reading [], Theorem ., [], Theorem ., [], Theorem , and [], Remark ., see also [, ]). For example, Mursaleen [] gives the following.

Theorem . Let <qn<pn≤such thatlimn→∞pn= andlimn→∞qn= .Then,for

each f ∈C[, ],Bn,pn,qn(f;x)converge uniformly to f on[, ].

All linear positive operators mentioned in the articles cited above require that lim_n→∞[n]pn,qn=∞; otherwise, these operators do not define approximation processes. However, the claim that bothlim_n→∞pn=  andlimn→∞qn=  with  <qn<pn≤ imply

thatlim_n→∞[n]pn,qn=∞, in general, is not true. A counterexample is presented below. Example . Letpn=  – /

√

n, then [n]pn,qn → for any sequence{qn}satisfying  <

qn<pn. Indeed,

≤[n]pn,qn=p

n–

n +pnn–qn+· · ·+pnqnn–+qnn–

Later, the author [] presented a more accurate assertion: Letqn,pnsuch that  <qn<

pn≤ andqn→,pn→,qnn→a,pnn→b(a<b) asn→ ∞, thenlimn→∞[n]pn,qn→ ∞. It is natural to ask: What is the class of sequences (pn,qn) satisfyinglimn→∞[n]pn,qn=∞ when limn→∞pn=  andlimn→∞qn=  under  <qn<pn≤? Undoubtedly, this is an

important problem. In this note, we will solve this problem in Section .

2 Main results

For  <qn<pn≤, setqn:=  –αn,pn:=  –βnsuch that ≤βn<αn< ,αn→,βn→

asn→ ∞. In the sequel, we use notationan∼bn(an,bn> )⇔limn→∞bann= . First, let us present the following auxiliary proposition.

Lemma . Let n∈N,then as n→ ∞we have

[n]pn,qn→ ∞ ⇒ e

nβn_/n→_{. (.)}

On the other hand,

enαn_/n→_ ⇒ _[n]

pn,qn→ ∞. (.)

Proof We note that

[n]pn,qn=q

n–

n +pnqnn–+· · ·+pnn–qn+pnn–=qnn–

 +pn/qn+· · ·+ (pn/qn)n–

>nqn_n–=n( –αn)(–/αn)(n–)(–αn)∼n/enαn, n→ ∞. (.)

Similarly, we have

[n]pn,qn=q

n–

n +pnqnn–+· · ·+pnn–qn+pnn–=pnn–

 +qn/pn+· · ·+ (qn/pn)n–

<npn_n–=n( –βn)(–/βn)(n–)(–βn)∼n/enβn, n→ ∞. (.)

Therefore, from (.) and (.), for sufficiently largen, there exist two positive real num-bersCandCsatisfying

C·n/enαn< [n]pn,qn<C·n/e

nβn_{. (.)}

This yields the proof.

The main result of this work is expressed by the next assertion. Theorem . The following statements are true:

(A) Iflimn→∞en(βn–αn)= andenβn/n→,then[n]pn,qn→ ∞.

(B) Iflim_n→∞en(βn–αn)_{< ande}nβn_(α

n–βn)→,then[n]pn,qn→ ∞.

(C) Iflim_n→∞en(βn–αn)_{< ,lim}

n→∞en(βn–αn)= andmax{enβn/n,enβn(αn–βn)} →,then

[n]pn,qn→ ∞. Conversely,

(B) Iflimn→∞en(βn–αn)< and[n]pn,qn→ ∞,thenen

βn_(α

n–βn)→.

(C) If lim_n→∞en(βn–αn) _{< ,lim}

n→∞en(βn–αn) =  and [n]pn,qn → ∞, then max{en

βn_/n,

enβn_(α

Proof Case (A):

If limn→∞en(βn–αn) = , sincelimn→∞enαn/n=limn→∞enβn/n/en(βn–αn) → and

com-bined with (.) imply [n]pn,qn→ ∞(see Remark .). Case (B) and Case (B):

Iflimn→∞en(βn–αn)< . Note that, for sufficiently largen,

[n]pn,qn=p

n–

n

 – (qn/pn)n

 –qn/pn ∼

 enβn

 – ( – (αn–βn)/( –βn))n

(αn–βn)/( –βn)

. (.)

Since

αn–βn

 –βn →

, (.)

it is not difficult to obtain from (.) andlimn→∞en(βn–αn)<  that, for sufficiently largen,

there existsc∈(, ) such that

 –αn–βn  –βn

n

∼en(βn–αn)_{<c. (.)}

Setdn:=  – ( –α_–n–ββnn)

n_{, then for sufficiently largen,d}

n>  –c. Thus, from (.), (.) and

(.), we have

[n]pn,qn∼  enβn ·

dn

αn–βn

, (.)

which entails that

[n]pn,qn→ ∞ ⇔ e

nβn_(α

n–βn)→. (.)

This yields the proof of Case (B) and Case (B). Case (C) and Case (C):

If lim_n→∞en(βn–αn)_{< ,lim}

n→∞en(βn–αn)= . Since the sequence  <xn:=en(βn–αn)< 

is bounded, set E := {x|xis a limit point of{xn},n ≥ }, then supE =  from

limn→∞en(βn–αn)= . Now, we are going to extract a subsequence{xnk}of{xn}such that

(a) limk→∞xnk= , and

(b) E_{:={x|xis a limit point of{x}

n} \ {xnk}}, withsupE

_{< .}

We verify that it is possible to extract such a subsequence{xnk}. Since  is a limit point ofA:={xn,n≥}, take a subsequence{x_n()

k }ofAsuch that

limk→∞x_n()

k = , setA

()_:=

{x

n()_k ,k≥}and letA:=A\A

()_{. If  is also a limit point ofA}

, take a subsequence{x_n() k }

of

Asuch thatlimk→∞x_n()

k = , setA

()_:={x

n()_k ,k≥}and letA:=A\A

()_{. Continuing this}

process, we obtain a series of sequences, i.e.,A(),A(), . . . , setAf:=A\

s∈I⊆NA(s). Since

Ais a countable set, this process will stop until  is not a limit point ofAf after finite or

countable steps; otherwise, we will see that  is the only limit point ofA, which contradicts to the assumptions of Case (C). Then we can take the subsequence{xnk}=

s∈NA(s)which

satisfies (a) and (b).

Set {xn_k}:={xn} \ {xnk}, it is obvious that {nk,k≥} ∪ {nk,k≥}=N. Then from

limn→∞xn_k <  from (b), we also have seen from (B) that [n]p_n

k,qnk → ∞ ⇔ en

kβn k(_α

n_k–βn_k)→ ask→ ∞. In summary,

(i) limn→∞[n]pn,qn=∞ ⇔for any subsequence{Nk}of a natural number set such that

limk→∞Nk=∞, we havelimk→∞[n]p_Nk,q_Nk=∞.

(ii) Since{nk}k≥and{n_k}k≥are two subsequences of a natural number set such that

limk→∞nk=∞andlimk→∞nk=∞, thus ask→ ∞

lim

n→∞[n]pn,qn=∞ ⇒

⎧ ⎨ ⎩

[n]p_nk,q_nk→ ∞ ⇔ enkβnk/nk→,

[n]p_n

k,qnk→ ∞ ⇔ e

n_kβ_n

k_(α

n_k–βn_k)→.

We assert that the inverse proposition of (ii) also holds. Indeed, for each sufficiently large N> , there exists a positive integerKsuch that for every natural numberk>K, we have

[n]p_nk,q_nk >N; meanwhile, for the previousN> , there exists a positive integerK such

that for every natural numberk>K, we have [n]p_n

k,qnk

>N; taken=max{nK,nK}, for every natural numbern>n, we have [n]pn,qn>N, i.e.,limn→∞[n]pn,qn=∞. Now we have proved that ask→ ∞

lim

n→∞[n]pn,qn=∞ ⇔ :=

⎧ ⎨ ⎩

enkβnk_/nk→_{, (c)} en

kβn_k (α_n

k–βnk)→. (d)

(.)

Next, we infer that ask→ ∞

⇔ lim

n→∞max

enβn_/n,enβn_(α

n–βn)

= . (.)

‘⇐’ of (.) is straightforward. Now we show ‘⇒’. On the one hand, by Remark (.),

we know from (d) ofthaten

kβn_k

/n_k→, and combined with (c)enkβnk_/nk→_{, we can} showenβn_/n→_{ by using a similar method as in the previous paragraph. On the other} hand,enkβnk_(αn

k–βnk)→ is straightforward (sincelimk→∞xnk=limk→∞e

nk(βnk–α_nk)

= ,

and note (c) in ), and combined withen

kβn_k (α_n

k –βnk)→, we can also deduce that enβn_(α

n–βn)→. This yields the proof of ‘⇒’ in (.).

Therefore, (C) and (C) follow from (.) and (.). Remark . In general, from (.) we have only [n]pn,qn→ ∞ ⇒en

βn_/n→_{. However, if}

lim_n→∞en(βn–αn)_{=  holds, then we have the equivalent relatione}nβn_/n→_⇔_[n]

pn,qn→ ∞from (A). Thus, we have:

(A) Iflimn→∞en(βn–αn)= , thenenβn/n→⇔[n]pn,qn→ ∞. Similarly, from (B) and (B), (C) and (C) we have

(B) Iflimn→∞en(βn–αn)< , thenenβn(αn–βn)→⇔[n]pn,qn→ ∞.

(C) Iflimn→∞en(βn–αn)< ,limn→∞en(βn–αn)= , thenmax{enβn/n,enβn(αn–βn)} →⇔

[n]pn,qn→ ∞.

Remark . In Case (B), we can also deduce directly from limn→∞en(βn–αn) <  and

enβn_(α

com-bined with the classical inequality on upper (lower) limit

lim

n→∞e

n(αn–βn)_{· lim}

n→∞e

n(βn–αn)_{≥ lim}

n→∞

en(αn–βn)_·en(βn–αn)_{= ,}

we havelim_n→∞en(αn–βn)_{> . Thus, for sufficiently largen, there existsc}

>  such that

en(αn–βn)_>c

. This means that  < (logc)enβn/n<enβn/n·(n(αn–βn))→, and we have

seen thatenβn_/n→_.

Remark . Now we utilize Theorem . (Remark .) to elaborate Example . again. In the example,βn= /

√

n, whileenβn_/n=e√n_{/n asn}→ ∞_{, thus [n]}

pn,qn∞.

Forpn= ,βn= ,qn=  –αn, thenenβn/n= /n→ andenβn(αn–βn) =αn→. Thus

any case of (A)-(C) is straightforward. In this case, (pn,qn)-integer reduces toqn-integer,

and it is known that [n]qn→ ∞ ⇔qn→ as n→ ∞. See [], Theorem , and [],

formula (.).

3 Conclusion

In this note, we mainly obtain the sufficient and necessary conditions for (p,q)-integer [n]p,q tending to infinity as n→ ∞. The conclusion guarantees the (p,q)-analogue of

Bernstein-type operators to be approximation processes asn→ ∞.

Acknowledgements

The paper has been greatly improved by the efforts of the anonymous referees through their diligent reading of and perceptive comments on an initial draft.

This work is supported by the National Natural Science Foundation of China (Grant No. 61572020,11601266), the China Postdoctoral Science Foundation funded project (Grant No. 2015M582036) and the Natural Science Foundation of Fujian Province of China (Grant No. 2016J05017). We also thank Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing of Fujian Province University.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

XWX carried out the proof of the main results. QBC read and approved the final manuscript.

Author details

1_{School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou, China.}2_{School of} Mathematical Sciences, Xiamen University, Xiamen, China.

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Received: 23 February 2017 Accepted: 1 June 2017 References

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