# Derivatives of MultivariateBernstein Operators and Smoothness with Jacobi Weights

Authors: Jianjun Wang; Zuoxiang Peng; Shukai Duan; Jia Jing
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Using the modulus of smoothness, directional derivatives of multivariateBernstein operators with weights are characterized. The obtained results partly generalize the corresponding ones for multivariate Bernstein operators without weights.


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## Body

## 1. Introduction

For the simplexS=Sd in Rd(d=1,2,…),(1.1)S={x=(x1,x2,…,xd);xi≥0,i=1,2,…,d,|x|=∑i=0dxi≤1},we denoteC(S) the space of continuous functions on S equipped with the norm(1.2)‖f‖=sup⁡x∈S|f(x)|.
Let f∈C(S), for each n∈N0 (N0=N∪{0},N0d=N0×N0×⋯×N0∈Rd), the multivariate Bernstein polynomial of f is defined by(1.3)Bn,d(f;x)=∑|k|≤nPn,k(x)f(kn),x∈S,
where k=(k1,k2,…,kd) with nonnegative integers ki(i=0,1,2,…,n), and(1.4)Pn,k(x)=n!(n-|k|)!xk(1-|x|)n-|k|,|x|=∑i=0dxi,|k|=∑i=0dki,with the convention(1.5)k!=k1!k2!⋯kd!,xk=x1k1x2k2⋯xdkd.Obviously, the multivariateBernstein operators given in (1.3) can be reduced as the classical Bernstein polynomials in case d=1, that is,(1.6)Bn(f,x):=Bn,1(f,x):=∑k=0nf(kn)Pn,k(x),x∈[0,1].Here introduce the crucial notations of our investigation. First, with the simplexS, we denote VS the set of unit vectors in the directions of the edges of S where ei and -ei are considered to be the same vectors. That is, ei=(0,0,…,1ith,0,…,0)(1≤i≤d) and eij=ei-ej(1≤i<j≤d). With a direction ξ∈VS and a point x∈S, we define the step-weight function(1.7)φξ2(x)=inf⁡x+λξ∉S,λ>0d(x,x+λξ)inf⁡x-λξ∉S,λ>0d(x,x-λξ),whered(x,y) is the Euclidean distance between x and y in Rd. Obviously, as x∈S, the φξ2(x) can further be expressed as:(1.8)φξ2(x)={xi(1-|x|),ifξ=ei,1≤i≤d,2xixjifξ=ei-ej2,1≤i<j≤d.It is clear that φξ2(x) can be reduced as the classical Bernstein polynomials’ step-weight function φ2(x)=φξ(x)2=x(1-x)(x∈[0,1]) in case d=1.The multivariateJacobi weight function in this paper is denoted as follows:(1.9)ω(x)=xα(1-|x|)β,x∈S,whereα=(α1,α2,…,αd)∈Rd, 0<αi,β<1,i=1,2,…,d,  xα=x1α1x2α2,…,xdαd.Therth symmetric difference of function f with the direction e is given by(1.10)Δherf(x)={∑i=0rCri(-1)if(x+(r2-i)he),ifx±rhe2∈S,0,otherwise.Using the above notation, the weighted Sobolev space inS is then defined by(1.11)Wϕr,∞(S)={f∈C(S):ωf∈C(S),f∈Cr(S°),ωφijrDijrf∈C(S),1≤i≤j≤d,r∈N},
where  S° is the inner of S.Furthermore, the weightedK-functional is defined by(1.12)Kφr(f,tr)ω=inf⁡g∈Wϕr,p{‖ω(f-g)‖+tr∑1≤i≤j≤d‖ωφijrDijrg‖},and the weighted modulus is(1.13)Ωφr(f,t)ω=sup⁡0<h≤t∑1≤i≤j≤d‖ωΔhφijeijrf‖,where∥ωf∥=maxx∈S|ω(x)f(x)|is the weighted form. From [1], there exists a positive constant C,(1.14)C-1Kφr(f,tr)ω≤Ωφr(f,t)ω≤CKφr(f,tr)ω.Throughout the paper, the letterC, appearing in various formulas, denotes a positive constant independent of n, x, and f. Its value may be different at different occurrences, even within the same formula.The close connection between the derivatives ofBernstein-type operators and the smoothness of functions has been well investigated by Ditzian, Totik, Ivanov and some other mathematicians (see [2–6], etc.) In [2], Ditzian has studied the relations between the derivatives of classical Bernstein operators Bn,1(f,x) and the smoothness of the function f. In [7], we have presented the relation between the derivatives of classical Bernstein operators and the smoothness of function f with Jacobiweights. Zhou has considered the approximation problems of higher-dimensional Bernstein operators with Jacobi weights, and has pointed out the unboundedness of Bernstein operators with Jacobi weights in the usual norm [8]. Because of the unboundedness of Bn,d(f,x) operators with weights in C(S), he used the method of space reduction, that is,(1.15)C0(S)={f∈C(S):f(x)|x∈∂S=0}has been taken instead ofC(S)(∂S is the boundary of S). He then has shown the characteristic of the two dimensional Bernstein operators with Jacobi weights. In [1], Cao has yielded the order of approximation of d-dimensional Bernstein Operators with Jacobi weights by using the equivalence relation (1.14). In [6], Cao has evaluated extensively derivatives of the multivariate Bernstein operators on a simplex, and he proved the following.Theorem 1.1.
Letf∈C(S),0<α≤r,0≤λ≤1,r∈N,andξ∈VS, and suppose Ωrξ(f,t)=O(tα),then
(1.16)‖φξrλ(∂∂ξ)rBn,d(f,x)‖=O{min⁡(n2-λ,nφξ2(1-λ))(r-α)/2}.In this paper, we study the characterization of derivatives of multivariateBernstein polynomials with Jacobi weights by using the measure of smoothness in the space C0(S). The main result is expressed as follows.Theorem 1.2.
Letf∈C0(S), 0<α≤r, r∈N, and ξ∈VS, and suppose Ωrξ(f,t)ω=O(tα), one has
(1.17)‖ωφξ2r(∂∂ξ)2rBn,d(f,x)‖=O(nr-α).Remark 1.3.
Theorem1.2 shows that the characterization of derivatives for multivariate bernstein operators with jacobi weight by using the measure of smoothness Ωrξ(f,t)ω. conversely, we conjecture that the inverse theorem is also correct, that is,
(1.18)‖ωφξ2r(∂∂ξ)2rMn,d(f,x)‖=O(nr-α)⟺Ωrξ(f,t)ω=O(tα).
The above equivalent relation without Jacobi weight has been proved in [6] when λ=1. In fact, the proof of Theorem 1.2 shows that the direct part holds true, we leave the inverse part as an open problem.


## 2. Lemmas

To prove Theorem1.2, some lemmas will be shown in this section.Lemma 2.1.
Consider the following;(2.1)∑|k|≤nPn,k(x)ω-1(kn)≤Cω-1(x).Proof.
Whend=1, one has
(2.2)∑k=0nPn,k(x){nk+1}α{nn-k+1}β≤[∑k=0nPn,k(x){nk+1}2α]1/2[∑k=0nPn,k(x){nn-k+1}2β]1/2=:I1/2J1/2.
Consider different conditions,(1) if0<2α<1,
(2.3)I≤{∑k=0nPn,k(x)nk+1}2α{∑k=0nPn,k(x)}1-2β≤Cx-2α,(2) if1<2α<2,let2α=1+r,0≤r<1,
(2.4)I=∑k=0nPn,k(x){nk+1}{nk+1}r≤2x∑k=0nPn+1,k+1(x){nk+1}r≤Cx-(1+r)=Cx-2α.By the same methods J≤C(1-x)-2β can also be given.
Suppose the lemma is correct whend-1. We prove the lemma is also correct when d. Through a simple computation, the following results can be easily obtained
(2.5)Pn,k(x)=Pn,k1(x1)Pn-k1,k̅(x̅1-x1),where k̅=(k2,k3,…,kd)x̅=(x2,x3,…,xd),
(2.6)ω(x)∑|k|≤nPn,k(x)ω-1(kn)=ω(x)∑|k|≤nPn,k1(x1)Pn-k1,k̅(x̅1-x1)(k1n)-α1(k2n)-α2⋯(kdn)-αd(1-|k|n)-β=x1α1∑k1=0nPn,k1(x1)x2α2x3α3⋯xdαd(1-|x|)β(k1n)-α1(n-k1n)-|α̅|-β×∑|k̅|≤n-k1Pn-k1,k̅(x̅1-x1)⋅(k2n-k1)-α2⋯(kdn-k1)-αd(1-|k̅|n-k1)-β=x1α1∑k1=0nPn,k1(x1)x2α2x3α3⋯xdαd(1-|x|)β(k1n)-α1(n-k1n)-|α̅|-β×∑|k̅|≤n-k1Pn-k1,k̅(x̅1-x1)ω-1(k̅n-k1)≤Cx1α1∑k1=0nPn,k1(x1)x2α2x3α3⋯xdαd(1-|x|)β(k1n)-α1(n-k1n)-|α̅|-βω-1(x̅1-x1)≤Cx1α1∑k1=0nPn,k1(x1)(1-x1)|α̅|+β(k1n)-α1(n-k1n)-|α̅|-β≤C.Lemma 2.2.
Letf∈C0(S), r∈N,andξ∈VS, then
(2.7)‖ωφξ(x)2r(∂∂ξ)2rBn,d(f,x)‖≤Cnr‖ωf‖f∈C0(S).Proof.
First, we recall the discussion of theorem 4.1 of [9] that will allow us to consider lemma 1 with ξ=e2. it is clear that if ξ=ei,i=1,3,4,…,d, we may just rename the coordinates. the following transformation will help us to complete the other case of ξ. the transformation T:S→S is defined by [9](2.8)T:{T(x1,x2,…,xd)=(u1,u2,…,ud),T2=I,
where ui=xi(i≠j); uj=1-|x| and I is the identity operator.
Obviously,(2.9)∂∂ui=∂∂xi-∂∂xj,i≠j,∂∂uj=-∂∂xj,(2.10)Bn,d(f;x)=Bn,d(fT;Tx),Bn,d(f;Tx)=Bn,d(fT;x),
where fT(u)=f(x) and u=Tx. So, for ξ=eij/2, 1≤i<j≤d, we have
(2.11)‖ωφξ2r(∂∂ξ)2rBn,d(f)‖=‖ωTφei2r(∂∂ui)rBn,d(fT)‖≤Cnr‖ωTfT‖≤Cnr‖ωf‖.Secondly, we prove
(2.12)‖ωφe22r(∂∂x2)2rBn,d(f)‖≤Cnr‖ωf‖.
In The following we use mathematical induction on the dimension number d to prove (2.12). When d=1, Lemma 3.2 in [10] proved the above inequality for r=1, for r>1, from the expression of derivatives of Bernstein operator in [4] (page125,(9.4.3)), we can easily prove it. Next, suppose that (2.12) is valid for d-1(d>1); we prove (2.12) is also true for d. Assume
(2.13)S′={x̅:(x1,x̅)∈Sd},x̅=(x2,x3,…,xd),k̅=(k2,k3…,kd),k=(k1,k̅).
Letz=x̅/(1-x1)=(x2/(1-x1),x3/(1-x1),…,xd/(1-x1)).ω(x) can therefore be rewritten as
(2.14)ω(x)=x1α1(1-x1)|α̅|+βω(z),
and Bn,d(f,x) can be decomposed as
(2.15)Bn,d(f,x)=∑k1=0npn,k1(x1)Bn-k1,d-1(H,z),where H(u)=f(k1/n,(1-k1/n)u). Using the inductive assumption, we have
(2.16)|ωφe22r(∂∂x2)2rBn,d(f)|=x1α1(1-x1)|α̅|+β∑k1=0npn,k1(x1)z1α2⋯zd-1αd(1-|z|)βφe12r(z)(∂∂z1)2rBn-k1,d-1(H,z)=x1α1(1-x1)|α̅|+β∑k1=0npn,k1(x1)ω(z)φe12r(z)(∂∂z1)2rBn-k1,d-1(H,z)≤x1α1(1-x1)|α̅|+β∑k1=0npn,k1(x1)C(n-k1)rmax⁡Z∈Sd-1|ω(z)H(z)|≤Cnr‖ωf‖x1α1(1-x1)|α̅|+β∑k1=0npn,k1(x1)(k1n)-α1(n-k1n)-|α̅|-β≤Cnr‖ωf‖.Here, the equality
(2.17)ω(z)H(z)=(k1n)-α1(n-k1n)-|α̅|-β(ωf)(k1n,(1-k1n)z),
and the inequality
(2.18)x1α1(1-x1)|α̅|+β∑k1=0npn,k1(x1)(k1n)-α1(n-k1n)-|α̅|-β≤C
have been used in the proof of (2.16). The proof of Lemma 2.2 is complete.Lemma 2.3.
Letr∈Nandξ∈VS, then
(2.19)‖ωφξ(x)2r(∂∂ξ)2rBn,d(f,x)‖≤C‖ωφξ2r(∂∂ξ)2rf‖f∈C0r(S).Proof.
By (2.10), for η=ei,u=Txandξ=eij/2, 1≤i<j≤d, we have(2.20)‖ωφξ2r(∂∂ξ)2rBn,d(f,x)‖=‖ωTφη2r(∂∂η)2rBn,d(fT,Tx)‖≤C‖ωTφη2r(∂∂η)2r(fT)(u)‖≤C‖ωφξ2r(∂∂ξ)2rf‖p.
Similar to the discussion in the proof of Lemma 2.2., we need only to prove the case of ξ=e2, that is,(2.21)‖ωφe22r(∂∂x2)2rBn,d(f,x)‖≤C‖ωφe22r(∂∂x2)2rf‖.
The steps to prove (2.21) are similar to those to prove the inequality (2.12). Hence, the proof of Lemma 2.3 is complete


## 3. Proof of Theorem

We will prove Theorem1.2 in the followings. For ξ=e2 and for all g∈Wϕr,∞(S), it follows from Lemmas 2.2 and 2.3 that(3.1)‖ωφe22r(∂∂x2)2rBn,d(f,x)‖≤‖ωφe22r(∂∂x2)2rBn,d(f-g,x)‖+‖ωφe22r(∂∂x1)2rBn,d(g,x)‖≤C{nr‖ω(f-g)‖+‖ωφe22r(∂∂x2)2rg‖}.From the definition ofK-functional and (1.14), we obtain(3.2)‖ωφe22r(∂∂x2)2rBn,d(f,x)‖≤CnrKre2(f;n-r)ω≤CnrΩre2(f,1n)ω≤Cnr-α.Similarly, the case ofξ=ei, i=1,3,4,…,d can also be proved. If ξ=((ei-ej)/2)1≤i<j≤d, it is not difficult to obtain(3.3)‖ωφξ2r(∂∂ξ)2rBn,d(f,x)‖=‖ωTφη2r(∂∂η)2rBn,d(fT,u)‖≤Cnr-α,
by assuming η=ei,u=Tx. The proof of Theorem 1.2 is complete.



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*Source: 346132-2012-02-20.xml*