Approximation of a Kind of New Type Bézier Operators Mei-Ying Ren1* and Xiao-Ming Zeng2 - DocsLib

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Ren and Zeng Journal of Inequalities and Applications (2015)2015:412 DOI 10.1186/s13660-015-0940-9

R E S E A R C H Open Access Approximation of a kind of new type Bézier operators Mei-Ying Ren1* and Xiao-Ming Zeng2

*Correspondence: [email protected] Abstract 1School of Mathematics and Computer Science, Wuyi University, In this paper, a kind of new type Bézier operators is introduced. The Korovkin type Wuyishan, 354300, China approximation theorem of these operators is investigated. The rates of convergence Full list of author information is of these operators are studied by means of modulus of continuity. Then, by using the available at the end of the article 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 any α >,andafunctionf deﬁned on [, ] as follows: n k α α B α(f ; x)= f J (x)–J (x) ,() n, n n,k n,k+ k= n n i n–i where Jn,n+(x)=,andJn,k(x)= i=k Pn,i(x), k =,,...,n, Pn,i(x)= i x ( – x) . Jn,k(x)is the Bézier basis function of degree n.

Obviously, when α =,Bn,α(f ; x) become the well-known Bernstein polynomials Bn(f ; x), n and for any x ∈ [, ], we have  = Jn,(x)>Jn,(x)>··· > Jn,n(x)=x , 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 operators, which are based on the Bézier basis function, have been introduced and studied (e.g.,see[–]). In , Ren [] introduced Bernstein type operators as follows:

n– Ln(f ; x)=f ()Pn,(x)+ Pn,k(x)Bn,k(f )+f ()Pn,n(x), () k= ∈ ∈ n k n–k where f C[, ], x [, ], Pn,k(x)= k x ( – x) , k = ,,...,n,andBn,k(f )=   nk– n(n–k)– · · B(nk,n(n–k))  t ( – t) f (t) dt, k =,...,n –,B( , ) is the beta function.

© 2015 Ren and Zeng. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Ren and Zeng Journal of Inequalities and Applications (2015)2015:412 Page 2 of 10

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

j Remark  For Ln(t ; x), j =,,,wehave

(i) Ln(; x)=;

(ii) Ln(t; x)=x; n(n –) n + (iii) L t; x = x + x. n n + n +

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:

n– (α) (α) (α) Ln,α(f ; x)=f ()Qn,(x)+ Qn,k(x)Bn,k(f )+f ()Qn,n(x), () k=

∈ ∈ α (α) α α where f C[, ], x [, ], >, Q n,k(x)=Jn,k(x)–Jn,k+(x), Jn,n+(x)=, Jn,k(x)= n n i n–i   nk– i=k Pn,i(x), k = ,,...,n, Pn,i(x)= i x ( – x) ,andBn,k(f )= B(nk,n(n–k))  t ( – t)n(n–k)–f (t) dt, k =,...,n –,B(·, ·) is the beta function.

It is clear that Ln,α(f ; x) are linear and positive on C[, ]. When α =,Ln,α(f ; x)become

the operators Ln(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 convergence 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, for f ∈ C[, ], we denote f = max{|f (x)| : x ∈ [, ]}. ω(f , δ)(δ >)de- notes the usual modulus of continuity of f ∈ C[, ].

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

Lemma  (see []) Let α >.We have

 n (i) lim Jα (x)=xuniformlyon[, ]; n→∞ n n,k k=  n x (ii) lim kJα (x)= uniformly on [, ]. n→∞ n n,k  k=

Lemma  Let α >.We have

(i) Ln,α(; x)=;

(ii) lim Ln,α(t; x)=xuniformlyon[, ]; n→∞   (iii) lim Ln,α t ; x = x uniformly on [, ]. n→∞ Ren and Zeng Journal of Inequalities and Applications (2015)2015:412 Page 3 of 10

Proof By simple calculation, we obtain B () = , B (t)= k , B (t)=  (k + k ). n,k n,k n n,k n+ n n (α) (i) Since k= Qn,k(x)=,by()wecangetLn,α(; x)=. (ii) By (), we have

n– (α) k (α) L α(t; x)= Q (x) + Q (x) n, n,k n n,n k=  n – n = Jα (x)–Jα (x) + ···+ Jα (x)–Jα (x) + Jα (x) n, n, n n,n– n,n n n,n n  n = Jα (x), n n,k k=

thus, by Lemma (i), we have limn→∞ Ln,α(t; x)=x uniformly on [, ]. (iii) By (), we have n–   (α)  k (α) L α t ; x = Q (x) k + + Q (x) n, n + n,k n n,n k=  n  = k –+ Jα (x) n + n n,k k=

  n  n  n = n · kJα (x)–n · Jα (x)+ Jα (x) , n + n n,k n n,k n n,k k= k= k=

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

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

≥ ≤ (α) ≤ αPn,k(x), α ;  Qn,k(x) α Pn,k(x),  < α <.

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

n β | |β α ≤ –α α  k – nx Pn,k(x) (n +) (A β ) n , α k=

where the constant As only depends on s.

Lemma  For α ≥ , we have  α  (i) L α (t – x) ; x ≤ · ; n,  n α  (ii) L α |t – x|; x ≤ · . n,  n

Proof (i) By (), Lemma  and Remark ,weobtain  Ln,α (t – x) ; x n–  (α) (α)   (α) = x Qn,(x)+ Qn,k(x)Bn,k (t – x) +(–x) Qn,n(x) k= Ren and Zeng Journal of Inequalities and Applications (2015)2015:412 Page 4 of 10

n–    ≤ α x Pn,(x)+ Pn,k(x)Bn,k (t – x) +(–x) Pn,n(x) k=  = αLn (t – x) ; x (n +)α = x( – x), () n +

 n(n+) max ≤ ≤ ∈ ≤ since  x  x( – x)=  ,andforanyn N,onecanget n+ , so we have

 α  L α (t – x) ; x ≤ · . n,  n

(ii) In view of Ln,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 α  L α |t – x|; x ≤ · . n,  n

Lemma  For <α <,we have

 –α (i) Ln,α (t – x) ; x ≤ Mαn ; – α (ii) Ln,α |t – x|; x ≤ Mα · n  .

Here the constant Mα only depends on α.

Proof (i) By () and Lemma ,weobtain

 Ln,α (t – x) ; x n–  (α) (α)   (α) = x Qn,(x)+ Qn,k(x)Bn,k (t – x) +(–x) Qn,n(x) k= n– ≤  α α   α x Pn,(x)+ Pn,k(x)Bn,k (t – x) +(–x) Pn,n(x) k= n  k k = Pα (x) k + –x + x n,k n + n n k=  n  n k k = (k – nx)Pα (x)+ Pα (x) –x + x n + n,k n + n,k n n k= k=

:= I + I.

≤ n(n+) –α α ≤ α –α By Lemma ,wehaveI  (n +) (A  ) (A  ) n , where the constant A  n + α α α only depends on α. Ren and Zeng Journal of Inequalities and Applications (2015)2015:412 Page 5 of 10

n α ≤ –α n α | k Using the Hölder inequality, we have k= Pn,k(x) (n +) [ k= Pn,k(x)] ,and n – k |≤ x n + x , so we have

α  n I ≤ (n +)–α P (x)  n + n,k k=  = (n +)–α ≤ n–α. n +

α Denote Mα =(A  ) +,thenwecanget α  –α Ln,α (t – x) ; x ≤ Mαn .

(ii) Since

 Ln,α |t – x|; x ≤ Ln,α(; x) Ln,α (t – x) ; x ,

thus, we get – α Ln,α |t – x|; x ≤ Mα · n  .

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

Proof By () and Lemma (i), we have Ln,α(f ; x) ≤f Ln,α(; x)=f .

3 Main results

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

Theorem  Let α >.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 ,wehave j limn→∞ Ln,α(ej; ·)–ej =forej(t)=t , j = , , . So, according to the well-known

Bohman-Korovkin theorem ([], p., Theorem .), we see that the sequence {Ln,α(f ; x)} converges to f uniformly on [, ] for any f ∈ C[, ].

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

∈ ∈ Theorem  Let f C[, ], x [, ]. Then (i) when α ≥ , we have L (f ; ·)–f ≤( + α )ω(f , √ ); n,α √ n – α (ii) when <α <, we have Ln,α(f ; ·)–f ≤( + Mα)ω(f , n  ).

Here the constant Mα only depends on α. Ren and Zeng Journal of Inequalities and Applications (2015)2015:412 Page 6 of 10

Proof (i) When α ≥ , by Lemma (i), we have Ln,α(f ; x)–f (x) n– ≤ (α) (α) (α) f () – f (x) Qn,(x)+ Qn,k(x)Bn,k f (t)–f (x) + f () – f (x) Qn,n(x) k= n– ≤ | | (α) (α) | | | | (α) ω f , –x Qn,(x)+ Qn,k(x)Bn,k ω f , t – x + ω f , –x Qn,n(x) k= √  ≤ + n|–x| ω f , √ Q(α) (x) n n, n– √ (α)  + Q (x)Bn,k + n|t – x| ω f , √ n,k n k= √  + + n|–x| ω f , √ Q(α)(x) n n,n  √  ≤ ω f , √ + nω f , √ Ln,α |t – x|; x , n n | |≤ α ω √ so, by Lemma (ii), we obtain Ln,α(f ; x)–f (x) ( +  ) (f , n ). The desired result follows immediately. (ii) When  < α < , by Lemma (i), we have Ln,α(f ; x)–f (x) n– ≤ | | (α) (α) | | | | (α) ω f , –x Qn,(x)+ Qn,k(x)Bn,k ω f , t – x + ω f , –x Qn,n(x) k= n– α α (α) (α) α α ≤  | | –   | | –  +n –x ω f , n Qn,(x)+ Qn,k(x)Bn,k +n t – x ω f , n k= α α  | | –  (α) + +n –x ω f , n Qn,n(x) – α α – α = ω f , n  + n  ω f , n  Ln,α |t – x|; x , √ – α so, by Lemma (ii), we obtain |Ln,α(f ; x)–f (x)|≤( + Mα)ω(f , n  ). The desired result follows immediately.

Theorem  Letf ∈ C[, ], x ∈ [, ]. Then (i) when α ≥ , we have α  α α Ln,α(f ; x)–f (x) ≤ f + ω f , √ + ; n n  n

(ii) when <α <, we have α –α – –α Ln,α(f ; x)–f (x) ≤ f Mαn + ω f , n  ( + Mα) Mαn ,

where the constant Mα only depends on α. Ren and Zeng Journal of Inequalities and Applications (2015)2015:412 Page 7 of 10

Proof Let f ∈ C[, ]. For any t, x ∈ [, ], δ >,wehave t f (t)–f (x)–f (x)(t – x) ≤ f (u)–f (x) du x

≤ ω f , |t – x| |t – x|

≤ ω f , 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 –  ≤ ω f , δ Ln,α |t – x|; x + δ Ln,α (t – x) ; x

≤ ω f , δ Ln,α(; x) –   + δ Ln,α (t – x) ; x Ln,α (t – x) ; x .

So we get Ln,α(f ; x)–f (x) ≤ f Ln,α |t – x|; x –   + ω f , δ +δ Ln,α (t – x) ; x Ln,α (t – x) ; x .()

α ≥ δ √ (i) When ,taking = n in (), by Lemma  and inequality (), we obtain the desired result. α (ii) When <α <,takingδ = n–  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.√ For f ∈ C[, ], ϕ(x)= x( – x),  ≤ λ ≤ , x ∈ [, ], let λ λ hϕ (x) hϕ (x) ωϕλ (f , t)= sup sup f x + – f x – ≤ λ   

be the Ditzian-Totik modulus of ﬁrst order, and let λ Kϕλ (f , t)= inf f – g + t ϕ g () g∈Wλ

λ be the corresponding K-functional, where Wλ = {f |f ∈ ACloc[, ], ϕ f < ∞, f < ∞}. It is well known that (see [])

≤ Kϕλ (f , t) Cωϕλ (f , t), ()

for some absolute constant C >. Now we state our next main result. Ren and Zeng Journal of Inequalities and Applications (2015)2015:412 Page 8 of 10

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

Proof Let g ∈ Wλ, by Lemma (i) and Lemma ,wehave 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) .()

t Since g(t)= x g (u) du + g(x), Ln,α(; x)=,so,wehave t Ln,α(g; x)–g(x) ≤ Ln,α g (u) du; x x t λ –λ ≤ ϕ g Ln,α ϕ (u) du; x .() x

By the Hölder inequality, we get

t t λ λ  λ ϕ– (u) du ≤ √ du |t – x|– ,() x x u( – u) √ √ also, in view of  ≤ u + –u <,≤ u ≤ , we have t t    √ du ≤ √ + √ du x u( – u) x u –u √ √ √ √ ≤  | 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 λ λ ϕ– (u) du ≤ Cϕ– (x)|t – x|,() x

also, by ()and(), we have

λ –λ Ln,α(g; x)–g(x) ≤ C ϕ g Ln,α ϕ (x)|t – x|; x λ –λ = C ϕ g ϕ (x)Ln,α |t – x|; x . () Ren and Zeng Journal of Inequalities and Applications (2015)2015:412 Page 9 of 10

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 ≤ x( – x) n + ϕ(x) ≤ C √ ,() n

so, by ()and(), we obtain

–λ λ ϕ (x) Ln,α(g; x)–g(x) ≤ C ϕ g √ , () n

thus, by ()and(), we have

–λ λ ϕ (x) Ln,α(f ; x)–f (x) ≤ f – g + C ϕ g √ .() n

Then, in view of (), (), and (), we obtain –λ –λ ϕ (x) ϕ (x) Ln,α(f ; x)–f (x) ≤ CK λ f , √ ≤ Cω λ f , √ , ϕ n ϕ n

where C is a positive constant, in diﬀerent places the value of C may be diﬀerent.

4Conclusions In this paper, a new kind of 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 the modulus of continuity. Then, by using the Ditzian-Totik modulus of smoothness, a direct theorem concerned with an approximation for these operators is obtained. Further, we can also study the inverse theorem and an equivalent theorem 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 1School of Mathematics and Computer Science, Wuyi University, Wuyishan, 354300, China. 2School 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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