The Voronovskaja theorem for the Bernstein-Schurer bivariate operatos is established

Dan Bărbosu

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 33 (2004) no. 1, pp. 19–24ictp.acad.ro/jnaatTHE VORONOVSKAJA THEOREMFOR BERNSTEIN-SCHURER BIVARIATE OPERATORSDAN BĂRBOSU∗Abstract. The Voronovskaja theorem for the Bernstein–Schurer bivariate op-eratos is established.MSC 2000. 41A10, 41A36, 41A63, 41A80.Keywords. Bernstein operator, Bernstein–Schurer operator, bivariate opera-tor, Voronovskaja theorem.1. PRELIMINARIESFor the classical Bernstein operator Bm : C([0, 1]) → C([0, 1]), defined forany function f ∈ C([0, 1]) by(1) (Bm f)(x) =m∑k=0pm,k(x) f(km),where(2) pm,k =(mk)xk (1− x)m−k,Voronovskaja E., proved (see [6]) the result expressed in the following result.Theorem 1. Let f ∈ C([0, 1]) be a function two times derivable in the pointx ∈ [0, 1]. Then, the equality(3) limm→∞m{(Bm f)(x)− f(x)} = x(1−x)2 f′′(x)holds.The above result is known as “the Voronovskaja theorem”.Let p be a non-negative integer. In 1962, Schurer F. (see [5]) introduced andstudied the operatorB̃m,p : C([0, 1 + p]) → C([0, 1]), defined for any functionf ∈ C([0, 1 + p]) by:(4)(B̃m,p f)(x) =m+p∑k=0p̃m,k(x) f(km),where(5) p̃m,k(x) =(m+pk)xk(1− x)m+p−k.∗North University of Baia Mare, Departament of Mathematics and Computer Science,Victoriei 76, 4800 Baia Mare, Romania, e-mail: dbarbosu@ubm.ro.20 Dan Bărbosu 2In our earlier paper [2] was proved the following Voronovskaja-type theoremfor the operator (4).Theorem 2. [2, Th. 2.1]. Let p be a non-negative integer. If the functionf ∈ C([0, 1+p]) is two times derivable in the point x ∈ [0, 1+p], the followingequality(6) limm→∞(m+ p){(B̃m,p f )(x)− f(x)}= p · f ′(x) + x(1−x)2 f′′(x)holds.Let p, q be two non-negative integers. Using the method of parametricextensions in the paper [3] we construct the bivariate operator of Bernstein–Schurer B̃m,n,p,q : C([0, 1 + p]× [0, 1 + q]) → C([0, 1]× [0, 1]), defined for anyfunction f ∈ C([0, 1 + p]× [0, 1 + q]) by(7)(B̃m,n,p,q f)(x, y) =m+p∑k=0n+q∑j=0p̃m,k(x)p̃n,j(y) f(km ,jn).We proved (see [3, Th. 2.4]) that the sequence{B̃m,n,p,q f}m,n∈N convergesto f , uniformly on [0, 1 + p] × [0, 1 + q], for any f ∈ C([0, 1 + p] × [0, 1 + q]).An estimation of the approximation order of f ∈ C([0, 1 + p] × [0, 1 + q])by B̃m,n,p,q f was also gived, using the first order modulus of smoothness forbivariate functions (see [3, Th. 2.5]).The aim of the present paper is to establish a Voronovskaja-type theoremfor the bivariate Bernstein–Schurer operators, i.e. to calculate the followinglimit(8) limm→∞{(B̃m,m,p,q f)(x, y)− f(x, y)},where f ∈ C2,2 ([0, 1 + p]× [0, 1 + q]).2. AUXILLIARY RESULTWe have to give some properties of the B̃m,p and B̃m,n,p,q operators.Lemma 3. [1]. For any fixed point x ∈ [0, 1 + p], there exists a positiveconstant M1(x, p), depending on x and p, such thatB̃m,p((t− x)4;x) ≤M1(x, p)n−2for all x ∈ N∗.Lemma 4. [3]. If (x, y) ∈ [0, 1 + p]× [0, 1 + q] and m,n ∈ N∗ the followingequalities hold(i) B̃m,n,p,q ( e00;x, y) = 1;(ii) B̃m,n,p,q (e10;x, y) = {1 + pm−1}x;(iii) B̃m,n,p,q (e01;x, y) = {1 + q n−1}y;3 The Voronovskaja theorem for Bernstein-Schurer bivariate operators 21(iv) B̃m,n,p,q (e22;x, y) = (m+ p)m−2{(m+ p)x2 + x(1− x)}++(n+ q)n−2{(n+ q)y2 + y(1− y)},where e00(t, τ) = 1, e10(t, τ) = t, e01(t, τ) = τ, e22(t, τ) = t2 + τ2 are the testfunctions.Lemma 5. Let (x0, y0) ∈ [0, 1 + p] × [0, 1 + q] be a fixed point and let ϕ ∈C ([0, 1 + p]× [0, 1 + q]) such that ϕ(x0, y0) = 0. Thenlimn→∞B̃m,m,p,q(ϕ;x0, y0) = 0.Proof. By the properties of function ϕ, for every ε > 0 there exists a positiveconstant δ = δ(ε) such that for |t− x0| < δ , |τ − y0| < δ the following holds(9) |ϕ(t, τ)| < ε4 .On the other hand, because f ∈ C ([0, 1 + p] × [0, 1 + q]), there exists apositive constant C1, such that(10) |ϕ(t, τ)| ≤ C1.Next, we can write:(11)∣∣∣B̃m,m,p,q(ϕ; x0, y0)∣∣∣ ≤ m+p∑k=0m+q∑j=0p̃m,k(x0)p̃m,j(y0)∣∣∣ϕ( km , jm)∣∣∣ .Let us to divide the set of sum’s indices in the following four classesI1 = {(k, j) : |k/m− x0| < δ and |j/m− y0| < δ} ;I2 = {(k, j) : |k/m− x0| < δ and |j/m− y0| ≥ δ} ;I3 = {(k, j) : |k/m− x0| ≥ δ and |j/m− y0| < δ} ;I4 = {(k, j) : |k/m− x0| ≥ δ and |j/m− y0| ≥ δ} .If we denote ωk,j(x0, y0) = p̃m,k(x)p̃m,j(y)∣∣ϕ ( km , jm)∣∣, we can write:∣∣∣B̃m,m,p,q(ϕ; x0, y0)∣∣∣ ≤ ∑(k,j)∈I1ωk,j(x0, y0) +∑(k,j)∈I2ωk,j(x0, y0)(12)+∑(k,j)∈I3ωk,j(x0, y0) +∑(k,j)∈I4ωk,j(x0, y0).Let S1, S2, S3, S4 be the sums of the right side of (12). For the sum S1,we get that there exists a natural number m1 = m1(ε) such thatS1 =∑(k,j)∈I2ωj,k(x0, y0) ≤m+p∑k=0m+p∑j=0p̃m,k(x0)p̃m,j(y0)∣∣ϕ( km , jm)∣∣< ε4B̃m,m,p,q(e00;x0, y0) <ε4for m > m1.22 Dan Bărbosu 4Because | jm − yo| ≥ δ implies δ−2( jm − y0)2 ≥ 1, for the sum S2, we getS2 =∑(k,j)∈I2ωj,k(x0, y0)≤ δ−2∑(k,j)∈I2( jm − y0)2p̃m,k(x0)p̃m,j(y0)∣∣ϕ( km , jm)∣∣≤M · δ−2∑(k,j)∈I2( jm − y0)2p̃m,k(x0)p̃m,j(y0) ≤≤M · δ−2m+p∑k=0m+p∑j=0( jm − y0)2p̃m,k(x0)p̃m,j(y0)= M · δ−2B̃m,m,p,q((τ − y0)2;x0, y0),where M = max {|ϕ(x, y)| : (x, y) ∈ [0, 1 + p]× [0, 1 + q]} .Taking into account Lemma 3, we get that there exists a positive constantM2(m, q, y0) such thatS2 ≤M2(m, q, y0) y0 (1− y0) ·m−1 δ−2.In the same way, applying Lemma 3 and Lemma 4, we obtainS3 ≤M3(m, q, x0)x0 (1− x0) ·m−1 δ−2andS4 ≤M4(m, p, q, x0, y0)x0 y0 (1− x0)(1− y0) ·m−2 δ−4.It follows that for given positive numbers ε , δ, Mi (i = 1, 2, 3) and (x0, y0) ∈[0, 1 + p]× [0, 1 + q] there exists the natural numbers m2, m3, m4 such thatS2 ≤M2(m, q, y0) y0 (1− y0) ·m−1 δ−2 < ε4 , for m > m2,S3 ≤M3(m, q, x0)x0 (1− x0) ·m−1 δ−2 < ε4 , for m > m3,S4 ≤M4(m, p, q, x0, y0)x0 y0 (1− x0)(1− y0) ·m−2 δ−4 < ε4 , for m >m4.Therefore, there exists the natural number N = max{m1,m2,m3,m4} suchthat for all natural numbers n > N we have S1 + S2 + S3 + S4 < ε.We can conclude that limm→∞B̃m,m,p,q(ϕ;x0, y0) = 0, and the proof ends. 3. THE MAIN RESULTUsing the auxiliary results from section 2, we can now prove the Voronovska-ja type theorem for the bivariate operators of Bernstein–Schurer.Let us to denote by C2,2 ([0, 1 + p] × [0, 1 + q]) the space of all functionsf ∈ C ([0, 1 + p]× [0, 1 + q]) with partial derivatives of first and second orderbelonging to the space C ([0, 1 + p]× [0, 1 + q]).5 The Voronovskaja theorem for Bernstein-Schurer bivariate operators 23Theorem 6. Let f ∈ C2,2 ([0, 1 + p] × [0, 1 + q]). Then, for all (x, y) ∈[0, 1 + p]× [0, 1 + q] we havelimm→∞m{B̃m,m,p,q(f ;x, y)− f(x, y)}== p f ′x(x, y) + q f ′y(x, y) + +12{x(1− x)f′′x2(x, y) + y(1− y)f′′y2(x, y)}.(13)Proof. Let (x0, y0) ∈ [0, 1 + p] × [0, 1 + q] be a fixed point. By making useof the Taylor formula for (t, τ) ∈ [0, 1 + p]× [0, 1 + q] with the Peano’s form ofthe remainder term, we havef(t, τ) =f(x0, y0) + (t− x) · f ′x(x0, y0) + (τ − y) · f ′y(x0, y0)+ 12{(t− x0)2f ′′x2(x0, y0) + 2(t− x0)(τ − y0)f′′xy(x0, y0)++ (τ − y0)2f ′′y2(x0, y0)}+ ϕ(t, τ ;x0, y0)√(t− x0)4 + (τ − y0)4,where ϕ(◦, ∗;x0, y0) ∈ C([0, 1 + p]× [0, 1 + q]) and limt→x0,τ→y0ϕ(t, τ ;x0, y0) = 0.Taking into account Lemma 3 we getB̃m,m,p,q(f(t, τ);x0, y0)− f(x0, y0) == p ·m−1xf ′x(x0, y0) + qm−1yf ′y(x0, y0)+ 12{(p2m−2x20 + (m+ p)m−2x0(1− x0))f ′′x20(x0, y0)+ 2pqm−2x0y0f ′′xy(x0, y0) + q2m−2y20+ (m+ q)m−2y0(1− y0))f ′′y20 (x0, y0)}+ B̃m,m,p,q(ϕ(t, τ ; x0, y0)√(t− x0)4 + (τ − y0)4 ;x0, y).(14)By Cauchy’s inequality, it follows∣∣∣∣B̃m,m,p,q (ϕ(t, τ ; x0, y0)√(t− x0)4 + (τ − y0)4 ;x0, y0)∣∣∣∣ ≤≤{B̃m,m,p,q(ϕ2(t, τ ; x0, y0))} 12{B̃m,m,p,q((t− x0)4 + (τ − y0)4;x0, y0)} 12 .From Lemma 5 it follows(15) limm→∞B̃m,m,p,q(ϕ2(t, τ ; x0, y0);x0, y0) = 0and from Lemma 4 we getB̃m,m,p,q((t− x0)4 + (τ − y0)4;x0, y0)== B̃m,p((t− x0)4;x0) + B̃m,q((τ − y0)4; y0)≤ {M1(x0, p) +M2(y0, q)}m−2,(16)where M1(x0, p) and M2(y0, q) are positive constants. Taking into account(15) and (16), by (14) we arrive to the desired result. 24 Dan Bărbosu 6REFERENCES[1] Bărbosu D., Some properties of the fundamental polynomials of Bernstein–Schurerr,“Proc. of ICAM 3”, Baia Mare–Borşa, Oct. 10–13, 2002, to appear.[2] Bărbosu D., The Voronovskaja theorem for the Bernstein–Schurer operators, “Proc.of ICAM 3”, Baia Mare–Borşa, Oct. 10–13, 2002, to appear.[3] Bărbosu D., Bernstein–Schurer bivariate operators, Rev. Anal. Numér. Théor.Approx., to appear.[4] Delvos J, Schempp W., Boolean Methods in Interpolation and Approximation, Long-man Scientific & Technical, 1989.[5] Schurer F., Linear positive operators in approximation theory, Math. Inst.Techn.Univ. Delft. Report, 1962.[6] Voronovskaja E., Determination de la forme asymptotique d’approximation desfunctions par les polynômes de M. Bernstein, C.R. Acad. Sci. URSS, pp. 79–85, 1932.Received by the editors: November 12, 2002.

The Voronovskaja theorem for Bernstein-Schurer bivariate operators

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The Voronovskaja theorem for Bernstein-Schurer bivariate operators

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