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

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 ).

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