Approximation of a kind of new type Bézier operators

R E S E A R C H

Open Access

Approximation of a kind of new type

Bézier operators

Mei-Ying Ren

_{and Xiao-Ming Zeng}

*_{Correspondence:} [email protected] 1_{School of Mathematics and} Computer Science, Wuyi University, Wuyishan, 354300, China Full list of author information is available at the end of the article

Abstract

In this paper, a kind of new type Bézier operators is introduced. The Korovkin type approximation theorem of these operators is investigated. The rates of convergence of these operators are studied by means of modulus of continuity. Then, by using the Ditzian-Totik modulus of smoothness, a direct theorem concerned with an

approximation for these operators is also obtained.

MSC: 41A10; 41A25; 41A36

Keywords: Bézier type operators; Korovich type approximation theorem; rate of convergence; direct theorem; modulus of smoothness

1 Introduction

In view of the Bézier basis function, which was introduced by Bézier [], in , Chang [] defined the generalized Bernstein-Bézier polynomials for anyα> , and a functionf defined on [, ] as follows:

Bn,α(f;x) = n

k= f

k n

J_nα_,k(x) –J_nα_,k+(x), ()

whereJn,n+(x) = , andJn,k(x) =ni=kPn,i(x),k= , , . . . ,n,Pn,i(x) =

n

i xi( –x)n–i.Jn,k(x) is

the Bézier basis function of degreen.

Obviously, whenα= ,Bn,α(f;x) become the well-known Bernstein polynomialsBn(f;x),

and for anyx∈[, ], we have  =Jn,(x) >Jn,(x) >· · ·>Jn,n(x) =xn,Jn,k(x) –Jn,k+(x) =

Pn,k(x).

During the last ten years, the Bézier basis function was extensively used for constructing various generalizations of many classical approximation processes. Some Bézier type op-erators, which are based on the Bézier basis function, have been introduced and studied (e.g., see [–]).

In , Ren [] introduced Bernstein type operators as follows:

Ln(f;x) =f()Pn,(x) + n–

k=

Pn,k(x)Bn,k(f) +f()Pn,n(x), ()

where f ∈ C[, ], x ∈ [, ], Pn,k(x) =

n

k xk( – x)n–k, k = , , . . . ,n, and Bn,k(f) =



B(nk,n(n–k)) 

tnk–( –t)n(n–k)–f(t)dt,k= , . . . ,n– ,B(·,·) is the beta function.

The moments of the operatorsLn(f;x) were obtained as follows (see []).

Remark  ForLn(tj;x),j= , , , we have

(i) Ln(;x) = ;

(ii) Ln(t;x) =x;

(iii) Ln

t;x =n(n– ) n_{+ } x

₊ n+  n_{+ }x.

In the present paper, we will study the Bézier variant of the Bernstein type operators Ln(f;x), which have been given by (). We introduce a new type of Bézier operators as

follows:

Ln,α(f;x) =f()Q(α)n,(x) +

n–

k=

Q(α)_n,k(x)Bn,k(f) +f()Q(α)n,n(x), ()

where f ∈C[, ], x∈ [, ], α > , Q(α)_n,k(x) =Jα

n,k(x) – J

α

n,k+(x), Jn,n+(x) = , Jn,k(x) = n

i=kPn,i(x), k = , , . . . ,n, Pn,i(x) =

n

i xi( –x)n–i, and Bn,k(f) = _B(nk,n_(n–k))



tnk–( – t)n(n–k)–f(t)dt,k= , . . . ,n– ,B(·,·) is the beta function.

It is clear thatLn,α(f;x) are linear and positive onC[, ]. Whenα= ,Ln,α(f;x) become

the operatorsLn(f;x).

The goal of this paper is to study the approximation properties of these operators with the help of the Korovkin type approximation theorem. We also estimate the rates of con-vergence of these operators by using a modulus of continuity. Then we obtain the direct theorem concerned with an approximation for these operators by means of the Ditzian-Totik modulus of smoothness.

In the paper, forf ∈C[, ], we denotef=max{|f(x)|:x∈[, ]}.ω(f,δ) (δ> ) de-notes the usual modulus of continuity off∈C[, ].

2 Some lemmas

Now, we give some lemmas, which are necessary to prove our results.

Lemma (see []) Letα> .We have

(i) lim

n→∞

 n

n

k=

J_nα_,k(x) =x uniformly on[, ];

(ii) lim

n→∞

 n

n

k=

kJ_nα_,k(x) = x 

 uniformly on[, ].

Lemma  Letα> .We have

(i) Ln,α(;x) = ;

(ii) lim

n→∞Ln,α(t;x) =x uniformly on[, ];

(iii) lim

n→∞Ln,α

Proof By simple calculation, we obtainBn,k() = ,Bn,k(t) =k_n,Bn,k(t) = _n_+(k+k_n).

(i) Sincen_k=Q(α)_n,k(x) = , by () we can getLn,α(;x) = .

(ii) By (), we have

Ln,α(t;x) = n–

k=

Q(α)_n,k(x)k n+Q

(α)

n,n(x)

=J_nα_,(x) –J_nα_,(x) n+· · ·+

J_nα_,n–(x) –J_nα_,n(x)n–  n +J

α

n,n(x)

n n =  n n k= Jα

n,k(x),

thus, by Lemma (i), we havelim_n→∞Ln,α(t;x) =xuniformly on [, ].

(iii) By (), we have

Ln,α

t;x =  n_{+ }

n–

k= Q(α)_n,k(x)

k+k

n

+Q(α)_n,n(x)

= 

n_{+ }

n

k=

k–  +  n

J_nα_,k(x)

= 

n_{+ }

n·  n

n

k=

kJ_nα_,k(x) –n· n

n

k=

J_nα_,k(x) + n

n

k= J_nα_,k(x)

,

thus, by Lemma , we havelimn→∞Ln,α(t;x) =xuniformly on [, ].

Lemma (see []) For x∈[, ],k= , , . . . ,n,we have

≤Q(α)_n,k(x)≤

αPn,k(x), α≥;

Pα

n,k(x),  <α< .

Lemma (see []) For <α< ,β> ,we have

n

k=

|k–nx|β_Pα

n,k(x)≤(n+ )–α(Aβ α)

α_nβ_,

where the constant Asonly depends on s.

Lemma  Forα≥,we have (i) Ln,α

(t–x);x ≤α  ·

 n;

(ii) Ln,α

|t–x|;x ≤ α  ·  n.

Proof (i) By (), Lemma  and Remark , we obtain

Ln,α

(t–x);x

=xQ(α)_n,(x) +

n–

k=

Q(α)_n,k(x)Bn,k

≤α

xPn,(x) + n–

k=

Pn,k(x)Bn,k

(t–x) + ( –x)Pn,n(x)

=αLn

(t–x);x

=(n+ )α

n_{+ } x( –x), ()

sincemax≤x≤x( –x) =_, and for anyn∈N, one can get n_n(n_++)≤, so we have

Ln,α

(t–x);x ≤α  ·

 n.

(ii) In view ofLn,α(;x) = , by the Cauchy-Schwarz inequality, we have

Ln,α

|t–x|;x ≤Ln,α(;x)

Ln,α

(t–x)_{;x ,}

thus, we get

Ln,α

|t–x|;x ≤

α  ·



n.

Lemma  For <α< ,we have

(i) Ln,α

(t–x);x ≤Mαn–α; (ii) Ln,α

|t–x|;x ≤Mα·n–α_.

Here the constant Mαonly depends onα.

Proof (i) By () and Lemma , we obtain

Ln,α

(t–x);x

=xQ(α)_n,(x) +

n–

k=

Q(α)_n,k(x)Bn,k

(t–x) + ( –x)Q(α)_n,n(x)

≤xP_nα_,(x) +

n–

k=

Pα_n,k(x)Bn,k

(t–x) + ( –x)P_nα_,n(x)

=

n

k= Pα_n,k(x)

 n_{+ }

k+k

n

– xk n+x



= 

n_{+ }

n

k=

(k–nx)Pα_n,k(x) +  n_{+ }

n

k= Pα_n,k(x)

k n– x

k n+x



:=I+I.

By Lemma , we haveI≤ n_n(n_++)(n+ )–α(A

α) α_≤(A



α)

α_n–α_{, where the constantA}



Using the Hölder inequality, we haven_k=Pα

n,k(x)≤(n+ )–α[

n

k=Pn,k(x)]α, and|_nk–

xk_n+x_{| ≤, so we have}

I≤ 

n_{+ }(n+ ) –α

_n

k= Pn,k(x)

α

= 

n_{+ }(n+ )

–α_≤n–α_.

DenoteMα= (A

α)

α_{+ , then we can get}

Ln,α

(t–x)_{;x ≤Mαn}–α_.

(ii) Since

Ln,α

|t–x|;x ≤Ln,α(;x)

Ln,α

(t–x)_{;x ,}

thus, we get

Ln,α

|t–x|;x ≤Mα·n–α_.

Lemma  For f ∈C[, ],x∈[, ]andα> ,we have Ln,α(f;x)≤ f.

Proof By () and Lemma (i), we have

Ln,α(f;x)≤ fLn,α(;x) =f.

3 Main results

First of all we give the following convergence theorem for the sequence{Ln,α(f;x)}.

Theorem  Letα> .Then the sequence{Ln,α(f;x)}converges to f uniformly on[, ]for

any f ∈C[, ].

Proof Since Ln,α(f;x) is bounded and positive on C[, ], and by Lemma , we have

limn→∞Ln,α(ej;·) –ej =  for ej(t) =tj, j= , , . So, according to the well-known

Bohman-Korovkin theorem ([], p., Theorem .), we see that the sequence{Ln,α(f;x)}

converges tof uniformly on [, ] for anyf ∈C[, ].

Next we estimate the rates of convergence of the sequence{Ln,α}by means of the mod-ulus of continuity.

Theorem  Let f ∈C[, ],x∈[, ].Then

(i) whenα≥,we haveLn,α(f;·) –f ≤( +

_α )ω(f,

 √

n);

(ii) when <α< ,we haveLn,α(f;·) –f ≤( +

√

Proof (i) Whenα≥, by Lemma (i), we have Ln,α(f;x) –f(x)

≤f() –f(x)Q(α)_n,(x) +

n–

k=

Q(α)_n,k(x)Bn,kf(t) –f(x) +f() –f(x)Q(α)n,n(x)

≤ωf,| –x| Q(α)_n,(x) +

n–

k=

Q(α)_n,k(x)Bn,k

ωf,|t–x| +ωf,| –x| Q(α)_n,n(x)

≤ +√n| –x| ω

f,√ n

Q(α)_n,(x)

+

n–

k=

Q(α)_n,k(x)Bn,k

 +√n|t–x| ω

f,√ n

+ +√n| –x| ω

f,√ n

Q(α)_n,n(x)

≤ω

f,√ n

+√nω

f,√

n

Ln,α

|t–x|;x ,

so, by Lemma (ii), we obtain|Ln,α(f;x) –f(x)| ≤( + _α

)ω(f,  √

n). The desired result

fol-lows immediately.

(ii) When  <α< , by Lemma (i), we have

Ln,α(f;x) –f(x)

≤ωf,| –x| Q(α)_n,(x) +

n–

k=

Q(α)_n,k(x)Bn,k

ωf,|t–x| +ωf,| –x| Q(α)_n,n(x)

≤ +nα|_{ –x}| _ωf,n–α _Q(α)

n,(x) +

n–

k=

Q(α)_n,k(x)Bn,k

 +nα|_t–x| _ωf,n–α

+ +nα|_{ –x}| _ωf,n–α _Q(α)

n,n(x)

=ωf,n–α _+nα_ωf,n–α _Ln,α|_t–x|_{;x ,}

so, by Lemma (ii), we obtain|Ln,α(f;x) –f(x)| ≤( +

√

Mα)ω(f,n–α_{). The desired result}

follows immediately.

Theorem  Letf ∈C[, ],x∈[, ].Then

(i) whenα≥,we have

_{Ln,α(f;x) –f(x)≤f} α n+ω

f ,√

n

 +

α 

α n;

(ii) when <α< ,we have

Ln,α(f;x) –f(x)≤f Mαn–α+ω

f ,n–α _{( +Mα)Mαn}–α_,

Proof Letf ∈C_{[, ]. For anyt,x∈[, ],δ> , we have} f(t) –f(x) –f (x)(t–x)≤

t

x

f (u) –f (x)du

≤ωf ,|t–x| |t–x|

≤ωf ,δ |t–x|+δ–(t–x) ,

hence, by the Cauchy-Schwarz inequality, we have

Ln,α

f(t) –f(x) –f (x)(t–x);x

≤ωf ,δ Ln,α

|t–x|;x +δ–Ln,α

(t–x);x

≤ωf ,δ Ln,α(;x)

+δ–

Ln,α

(t–x)_{;x L}

n,α

(t–x)_{;x .}

So we get

Ln,α(f;x) –f(x)

≤f Ln,α

|t–x|;x +ωf ,δ  +δ–

Ln,α

(t–x)_{;x L}

n,α

(t–x)_{;x . ()}

(i) Whenα≥, takingδ=√

nin (), by Lemma  and inequality (), we obtain the

desired result.

(ii) When <α< , takingδ=n–α _{in (), by Lemma  and inequality (), we obtain the}

desired result.

Finally we study the direct theorem concerned with an approximation for the sequence

{Ln,α}by means of the Ditzian-Totik modulus of smoothness. For the next theorem we shall use some notations.

Forf ∈C[, ],ϕ(x) =√x( –x), ≤λ≤,x∈[, ], let

ωϕλ(f,t) = sup <h≤t

sup

x±hϕλ(x)  ∈[,]

f

x+hϕ

λ_(x) 

–f

x–hϕ

λ_(x) 

be the Ditzian-Totik modulus of first order, and let

Kϕλ(f,t) = inf

g∈Wλ

f –g+tϕλg ()

be the correspondingK-functional, whereWλ={f|f∈ACloc[, ],ϕλ_{f <∞,f <∞}.} It is well known that (see [])

Kϕλ(f,t)≤Cωϕλ(f,t), ()

Theorem  Let f ∈C[, ],α≥,ϕ(x) =√x( –x),x∈[, ], ≤λ≤.Then there exists an absolute constant C> such that

Ln,α(f;x) –f(x)≤Cωϕλ

f,ϕ –λ_(x)

√

n

.

Proof Letg∈Wλ, by Lemma (i) and Lemma , we have Ln,α(f;x) –f(x)

≤Ln,α(f–g;x)+f(x) –g(x)+Ln,α(g;x) –g(x)

≤f–g+Ln,α(g;x) –g(x). ()

Sinceg(t) =_xtg(u)du+g(x),Ln,α(;x) = , so, we have

Ln,α(g;x) –g(x)≤

Ln,α

t

x

g(u)du;x

≤ϕλgLn,α

_xtϕ–λ(u)du;x

. ()

By the Hölder inequality, we get

t

x

ϕ–λ(u)du≤ t

x



√

u( –u)du

λ|t–x|–λ, ()

also, in view of ≤√u+√ –u< , ≤u≤, we have

t

x



√

u( –u)du ≤

t

x



√

u+ 

√

 –u

du

≤|√t–√x|+|√ –x–√ –t|

≤ _|

t–x|

√

t+√x+

|t–x|

√

 –t+√ –x

≤|t–x|



√

x+ 

√

 –x

≤|t–x|ϕ–(x), ()

thus, by () and (), we obtain

t

x

ϕ–λ(u)du≤Cϕ–λ(x)|t–x|, ()

also, by () and (), we have

Ln,α(g;x) –g(x)≤CϕλgLn,α

ϕ–λ(x)|t–x|;x =Cϕλgϕ–λ(x)Ln,α

In view of () and Lemma (i), by the Cauchy-Schwarz inequality, we have

Ln,α

|t–x|;x ≤Ln,α(;x)

Ln,α

(t–x)_;x

≤

(n+ )α

n_{+ } x( –x)

≤Cϕ(x)√

n, ()

so, by () and (), we obtain

Ln,α(g;x) –g(x)≤Cϕλg

ϕ–λ(x)

√

n , ()

thus, by () and (), we have

Ln,α(f;x) –f(x)≤f –g+Cϕλg

ϕ–λ_(x)

√

n . ()

Then, in view of (), (), and (), we obtain

_Ln,α(f;x) –f(x)≤CKϕλ

f,ϕ –λ_(x)

√

n

≤Cωϕλ

f,ϕ –λ_(x)

√

n

,

whereCis a positive constant, in different places the value ofCmay be different.

4 Conclusions

In this paper, a new kind of type Bézier operators is introduced. The Korovkin type approx-imation theorem of these operators is investigated. The rates of convergence of these oper-ators are studied by means of the modulus of continuity. Then, by using the Ditzian-Totik modulus of smoothness, a direct theorem concerned with an approximation for these op-erators is obtained. Further, we can also study the inverse theorem and an equivalent the-orem concerned with an approximation for these operators.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Author details

1_{School of Mathematics and Computer Science, Wuyi University, Wuyishan, 354300, China.}2_{School of Mathematical} Sciences, Xiamen University, Xiamen, 361005, China.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 61572020), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant No. JA12324), and the Natural Science Foundation of Fujian Province of China (Grant Nos. 2014J01021 and 2013J01017).

Received: 13 September 2015 Accepted: 9 December 2015

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