# Approximation Order for Multivariate

Authors: Jianjun Wang; Chan-Yun Yang; Shukai Duan
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Using the equivalence relation betweenK-functional and modulus of smoothness, we
establish a strong direct theorem and an inverse theorem of weak type for multivariate
Bernstein-Durrmeyer operators with Jacobi weights on a simplex in this paper. We
also obtain a characterization for multivariate Bernstein-Durrmeyer operators with Jacobi
weights on a simplex. The obtained results not only generalize the corresponding ones for
Bernstein-Durrmeyer operators, but also give approximation order of Bernstein-Durrmeyer
operators.


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## Body

## 1. Introduction

LetS=Sd(d=1,2,…) be a simplex in Rd defined by (1.1)S={x=(x1,x2,…,xd):xi≥0,i=1,2,…,d,|x|=∑i=0dxi≤1}.
For p≥1, we denote by Lp(S) the space of p-order Lebesgue integrable functions on S with(1.2)‖ωf‖p={(∫S|ω(x)f(x)|pdx)1/p<∞1≤p<+∞,max⁡x∈S⁡|ω(x)f(x)|p=+∞,
where L∞(S)=C(S) denote the space of continuous functions on S. For f∈L(S), the multivariate Bernstein-Durrmeyer Operators with d variables on S are given by (1.3)Mn,d(f;x)=∑|k|≤nPn⋅k(x)(n+d)!n!∫SPn,k(u)f(u)du,
where Pn,k(x)=(n!/(k!(n-|k|)!))xk(1-|x|)n-|k|(x∈S) and x=(x1,x2,…,xd)∈Rd, k=(k1,k2,…,kd)∈N0d, with the convention (1.4)|x|=∑i=1dxi,xk=x1k1x2k2⋯xdkd,|k|=∑i=1dki,k!=k1!k2!⋯kd!.For multivariate Jacobi weightsω(x)=xα(1-|x|)β, (x∈S, α=(α1,…,αd)∈Rd, 0<αi, β<1, i=1,2,…,d, xα=x1α1x2α2⋯xdαd). We give some further notations, for x∈S, and we write φi(x)=φii(x)=xi(1-|x|)(1≤i≤d), φij(x)=xixj, (1≤i<j≤d) and (1.5)Di=Dii=∂∂xi,1≤i≤d,Dij=Di-Dj,1≤i<j≤d,Dijr=Dij(Dijr-1),1≤i≤j≤d,r∈N,Dk=D1k1D2k2⋯Ddkd,k∈N0d.Forf∈Lp(S), the weighted Sobolev space is given by(1.6)Wϕr,p(S)={f∈Lp(S):ωf∈Lp(S),Dkf∈Lloc⁡(S0),ωφijrDijrf∈Lp(S),|k|≤r,1≤i≤j≤d,r∈N},Wϕr,∞(S)={f∈C(S):ωf∈C(S),f∈Cr(S0),ωφijrDijrf∈C(S),1≤i≤j≤d,r∈N},
where S0 is the interior of S. To characterize the approximation capability of multivariate Bernstein-Durrmeyer operators, we introduce the weighted K-functional (1.7)Kφr(f,tr)ω=inf⁡g∈Wϕr,p{‖ω(f-g)‖p+tr∑1≤i≤j≤d‖ωφijrDijrg‖p}
and a measure of smoothness of f(1.8)ωφr(f,t)ω=sup⁡0<h≤t∑1≤i≤j≤d‖ωΔhφijeijrf‖p.Since 1967, Durrmeyer introduced Bernstein-Durrmeyer operators, and there are many papers which studied theirproperties [1–7]. In 1991, Zhang studied the characterization of convergence for Mn,1(f;x) with Jacobi weights. In 1992, Zhou [5] considered multivariate Bernstein-Durrmeyer operators Mn,d(f;x) and obtained a characterization of convergence. In 2002, Xuan et al. studied the equivalent characterization of convergence for Mn,d(f;x) with Jacobi weights and obtained the following result.Theorem 1.1.
Forωf∈Lp(S), 0<r<1, the following results are equivalent:(i)
∥ω(Mn,df-f)∥p=O(n-r);(ii)
Kφ2(f,t)ω=O(tr).In this paper, using the Ditzian-Totik modulus of smoothness, we will give the upper bound and lower bound of approximation function byMn,d(f;x) on simplex. The main results are as follows.Theorem 1.2.
Ifωf∈Lp(S), then
(1.9)‖ω(Mn,df-f)‖p≤C{ωφ2(f,1n)ω+‖ωf‖pn}.
And there exists a positive number δ(0<δ<1) such that the following inequality is satisfied:
(1.10)ωφ2(f,1n)ω≤Cn∑k=1n(nk)δ‖ω(Mn,df-f)‖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.From Theorem1.2, we can easily obtain the following corollary.Corollary 1.3.
Ifωf∈Lp(S), 0<r<1, we has the following equivalent results:(i)
∥ω(Mn,df-f)∥p=O(n-r);(ii)
Kφ2(f,t)ω=O(tr);(iii)
ωφ2(f,t)ω=O(t2r).


## 2. Some Lemmas

To prove Theorem1.2, we will show some lemmas in this section. For the simplex S, the transformation T: S→S[10] defined by (2.1)T(x1,x2,…,xd)=(u1,u2,…,ud),ul={xjl=j,1-|x|l≠j
satisfies T2=I, and I is the identity operator. So we have (2.2)∂∂ul=∂∂xl-∂∂xj(l≠j),∂∂uj=-∂∂xj,Mn,d(f;x)=Mn,d(fT;Tx);Mn,d(f;Tx)=Mn,d(fT;x),
where fT(u)=f(Tx).Lemma 2.1.
Ifωf∈Lp(S), then
(2.3)‖ωMn,df‖p≤‖ωf‖p,‖ω(Mn,df-f)‖p≤Cn(‖ωf‖p+∑1≤i≤j≤d‖ωφij2Dij2f‖p),f∈Wϕr,p(S).Proof.
LettingS′={x̅:(x1,x̅)∈Sd}, x̅=(x2,x3,…,xd), k̅=(k2,k3…,kd), k=(k1,k̅), Pn,k1(x1)=(n!/k1!(n-k1)!)x1k1(1-x1)n-k1, then
(2.4)Mn,d(f;x)=∑k1=0nPn,k1(x1)∑|k̅|≤n-k1Pn-k1,k̅(x̅1-x1)(n+d)!n!×∫01Pn,k1(u1)∫S′Pn-k1,k̅(u̅1-u1)f(u)du̅du1=∑k1=0nPn,k1(x1)(n+d)!n!∫01Pn,k1(u1)(1-u1)d-1∑|k̅|≤n-k1Pn-k1,k̅(x̅1-x1)×∫Sd-1Pn-k1,k̅(t)f(u1,(1-u1)t)dtdu1=∑k1=0nPn,k1(x1)(n+d)∫01Pn+d-1,k1(u1)Mn-k1,d-1(f(u1,(1-u1)⋅);x̅1-x1)du1.
Using the transformationT, (2.2), (2.4), the method of [7], we can easily get (2.3).Lemma 2.2 (see [8]).
Iff∈Lp(S), then
(2.5)C-1ωφr(f,t)ω≤Kφr(f,tr)ω≤Cωφr(f,t)ω.Proof.
Lemma2.2 is proved when f∈C(S) in [8]. Similarly, we can prove f∈Lp(S).Lemma 2.3.
If0<a<1, b>0, x∈(0,1), Pn,k(x)=Cnkxk(1-x)n-k is basis function of the classical Bernstein operators, then
(2.6)∑k=1n-1Pn,k(x)(nk)a≤Cx-a,∑k=1n-1Pn,k(x)(nn-k)b≤C(1-x)-b.Proof.
The first inequality can be inferred by Hölder inequality. In the following we prove the second inequality.(i)
If0<b<1, using Hölder inequality, we can easily obtain the result.(ii)
Ifb≥1, let b=m+r, m∈N, 0≤r<1, then
(2.7)∑k=1n-1Pn,k(x)(nn-k)b=∑k=1n-1Pn,k(x)(nn-k)m(nn-k)r≤C(1-x)-m∑k=1n-1Pn+m,k(x)(n+mn+m-k)r≤C(1-x)-m-r=C(1-x)-b.Lemma 2.3 is completed.Lemma 2.4.
Iff∈Lp(S), 1≤p≤∞, then
(2.8)‖ωφij2Dij2Mn,df‖p≤Cn‖ωf‖p1≤i≤j≤d.Proof.
In the following we use the induction on the dimension numberd to prove the result. The case d=1 was proved by Lemma  4 of [6]. Next, suppose that Lemma 2.4 is valid for d=r(r≥1); we prove it is also true for d=r+1. To observe this, we use a decomposition formula (2.4), and we have
(2.9)ω(x)φ222(x)D222Mn,d(f;x)=x1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)z1α2z2α3⋯zd-1αd×(1-|z|)βφ112(z)∫01Pn+d-1,k1(u1)D112Mn-k1,d-1(f(u1,(1-u1)⋅);z)du1,
where z=(z1,z2,…,zd-1)=(x2/(1-x1),x3/(1-x1),…,xd/(1-x1)). Thus we have
(2.10)∫S|ω(x)φ222(x)D222Mn,d(f;x)|ds≤C∫01x1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)∫01Pn+d-1,k1(u1)(n-k1)×∫z∈Sd-1|ω(z)f(u1,(1-u1)z)|dzdx1du1≤Cn+dn+1n∫01∑k1=0n(k1+1n+1)α1(n-k1+1n+1)|α-|+βPn+d-1,k1(u1)×∫z∈Sd-1|ω(z)f(u1,(1-u1)z)|dzdu1≤Cn∫01u1α1(1-u1)|α-|+β(1u1)α1(1-u1)-|α-|-β∫z∈Sd-1|(ωf)(u1,(1-u1)z)|dzdu1=Cn‖ωf‖1.
In the above derivation, we have used the formula [6]
(2.11)∫01x1α1(1-x1)|α-|+βPn,k1(x1)dx1≤C1n+1(k1+1n+1)α1(n-k1+1n+1)|α-|+β
and the inequality
(2.12)∑k1=0n(k1+1n+1)α1(n-k1+1n+1)|α-|+βPn+d-1,k1(u1)≤Cu1α1(1-u1)|α-|+β.
When p=∞, we have
(2.13)ω(x)φ222(x)D222Mn,d(f;x)=x1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)z1α2z2α3⋯zd-1αd×(1-|z|)βφ112(z)∫01Pn+d-1,k1(u1)D112Mn-k1,d-1(f(u1,(1-u1)⋅);z)du1,
where z=(z1,z2,…,zd-1)=(x2/(1-x1),x3/(1-x1),…,xd/(1-x1)).
From the Cauchy-Swartz inequality, Hölder inequality, and Lemma2.3, we have
(2.14)|ω(x)φ222(x)D222Mn,d(f;x)|≤Cx1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)∫01Pn+d-1,k1(u1)(n-k1)×max⁡z∈Sd-1|z1α2z2α3⋯zd-1αd(1-|z|)βf(u1,(1-u1)z)|du1≤Cn‖ωf‖∞x1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)∫01Pn+d-1,k1(u1)(1u1)α1(1-u1)-|α-|-βdu1≤Cn‖ωf‖∞x1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)(∫01Pn+d-1,k1(u1)u1-2α1du1)1/2×(∫01Pn+d-1,k1(u1)(1-u1)-2|α-|-2βdu1)1/2≤Cn‖ωf‖∞x1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)(∫01Pn+d-1,k1(u1)u1-2du1)α1/2×(∫01Pn+d-1,k1(u1)du1)(1-α1)/2(∫01Pn+d-1,k1(u1)(1-u1)-2ddu1)(|α-|+β)/2d×(∫01Pn+d-1,k1(u1)du1)1/2-(|α-|+β)/2d≤Cn‖ωf‖∞x1α1(1-x1)|α-|+β∑k1=2nPn,k1(x1)(n+d)(n+d-1k1(k1-1))α1/2×((n+d-1)!(n-d)!(n-d-k1-1)!(n+d-k1-1)!)(|α-|+β)/2d((n+d)1-(|α-|+β)/2d-α1/2)-1≤Cn‖ωf‖∞x1α1(1-x1)|α-|+β∑k1=1n-1Pn,k1(x1)(nk1)α1(nn-k1)|α-|+β≤Cn‖ωf‖∞.
By Riesz interpolation theorem, we get
(2.15)‖ωφ222D222Mn,df‖p≤Cn‖ωf‖p.
Similarly, the other cases for i=1,3,4,…,d(=j) can be proved. For i≠j, by the transformation T, we have
(2.16)‖ωφij2Dij2Mn,df‖p=‖ωTφjj2Djj2Mn,dfT‖p≤Cn‖ωTfT‖p=Cn‖ωf‖p.
Lemma 2.4 is completed.Lemma 2.5.
Iff∈Wϕr,p(S)⊂Lp(S), 1≤p≤∞, then
(2.17)‖ωφij2Dij2Mn,df‖p≤C‖ωφij2Dij2f‖p1≤i≤j≤d.Proof.
We use the induction on the dimension numberd to prove Lemma 2.5. The case d=1 was proved by Lemma  3 of [6], that is,
(2.18)‖ωφ2D2Mn,1f‖p≤C‖ωφ2D2f‖p.
Next, suppose that Lemma 2.5 is valid for d=r(r≥1), and we prove it is also true for d=r+1. Noticing formula (2.4), we have
(2.19)ω(x)φ222(x)D222Mn,d(f;x)=x1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)z1α2z2α3⋯zd-1αd×(1-|z|)βφ112(z)∫01Pn+d-1,k1(u1)D112Mn-k1,d-1(f(u1,(1-u1)⋅);z)du1,
where z=(z1,z2,…,zd-1)=(x2/(1-x1),x3/(1-x1),…,xd/(1-x1)). When p=1, from the inductive assumption of p=1, we have
(2.20)∫S|ω(x)φ222(x)D222Mn,d(f;x)|ds≤C∫01x1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)∫01Pn+d-1,k1(u1)×∫z∈Sd-1|ω(z)φ112(z)D112f(u1,(1-u1)z)|dzdx1du1≤Cn+dn+1∫01∑k1=0n(k1+1n+1)α1(n-k1+1n+1)|α-|+βPn+d-1,k1(u1)×∫z∈Sd-1|ω(z)φ112(z)D112f(u1,(1-u1)z)|dzdu1≤C∫01u1α1(1-u1)|α-|+β(1u1)α1(1-u1)-|α-|-β∫z∈Sd-1|(ωφ222D222f)(u1,(1-u1)z)|dzdu1≤C‖ωφ222D222f‖1.
When p=∞, we have
(2.21)ω(x)φ222(x)D222Mn,d(f;x)=x1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)z1α2z2α3⋯zd-1αd×(1-|z|)βφ112(z)∫01Pn+d-1,k1(u1)D112Mn-k1,d-1(f(u1,(1-u1)⋅);z)du1,
where z=(z1,z2,…,zd-1)=(x2/(1-x1),x3/(1-x1),…,xd/(1-x1)). From the inductive assumption, the Cauchy-Swartz inequality, Holder inequality, and Lemma 2.4, we get
(2.22)|ω(x)φ222(x)D222Mn,d(f;x)|≤Cx1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)∫01Pn+d-1,k1(u1)×max⁡z∈Sd-1|z1α2z2α3⋯zd-1αd(1-|z|)βφz2Dz2f(u1,(1-u1)z)|du1≤Cx1α1(1-x1)|α-|+β∑k1=0nPn,k1(x1)(n+d)∫01Pn+d-1,k1(u1)x1-α1(1-x1)-|α-|-β×‖ωφ222D222f‖∞du1≤C‖ωφ222D222f‖∞.
By Riesz interpolation theorem, we get
(2.23)‖ωφ222D222Mn,df‖p≤C‖ωφ222D222f‖p.
Similarly, the other cases for i=1,3,4,…,d(=j) can be proved. For i≠j, by the transformation T, we have
(2.24)‖ωφij2Dij2Mn,df‖p=‖ωTφjj2Djj2Mn,dfT‖p≤C‖ωTφjj2Djj2fT‖p≤C‖ωφij2Dij2f‖p
Lemma 2.5 is completed.Lemma 2.6 (see [9]).
Let{σn}, {ϕn} be nonnegative sequences (σ1=0, n∈N). For l>0, if the sequences {σn}, {ϕn} satisfy
(2.25)σn≤Q(kn)lσk+ϕk(Q≥1,1≤k≤n,n∈N),
one has
(2.26)σn≤Mn-s∑k=1nks-1ϕk.
If Q=1, then l=s. If Q>1, then 0<s<l.


## 3. The Proof of Theorems

Now we prove (1.9) of Theorem 1.2. By using Lemma 2.1, for arbitrary g∈Wϕr,p(S)⊂Lp(S), we have (3.1)‖ω(Mn,df-f)‖p≤C(‖ωMn,d(f-g)‖p+‖ωMn,dg-ωg‖p+‖ω(f-g)‖p)≤C(‖ω(f-g)‖p+1n(∑1≤i≤j≤d‖ωφij2Dij2g‖p+‖ωg‖p))≤C(‖ω(f-g)‖p+1n∑1≤i≤j≤d‖ωφij2Dij2g‖p+1n‖ωf‖p).
Hence, from Lemma 2.2, we obtain (3.2)‖ω(Mn,df-f)‖p≤C(Kφ2(f,1n)ω+1n‖ωf‖p)≤C(ωφ2(f,t)ω+1n‖ωf‖p).
Next, we prove (1.10) of Theorem 1.2. Leting σn=C(1/n)∥ωφij2Dij2Mn,d(f)∥p (1≤i≤j≤d), ϕn=C∥ω(Mn,d(f)-f)∥p, then σ1=0. By Lemmas 2.4 and 2.5, we have (3.3)σn≤C1n‖ωφij2Dij2Mn,d(f-Mk,df)‖p+C1n‖ωφij2Dij2Mn,dMk,df‖p≤C‖ω(f-Mk,df)‖p+C1n‖ωφij2Dij2Mk,df‖p=Cknσk+ϕk(C>1).
Using Lemma 2.6, we get σn≤C(1/n)∑k=1n(n/k)δϕk(0<δ<1). That is, (3.4)‖ωφij2Dij2Mn,d(f)‖p≤C∑k=1n(nk)δ‖ω(Mk,df-f)‖p.
When n≥2, there exists (m∈N) such that n/2≤m≤n and satisfies the equation (3.5)‖ω(Mm,df-f)‖p=min⁡n/2≤k≤n‖ω(Mk,df-f)‖p.
Thus, (3.6)‖ω(Mm,df-f)‖p≤2n∑n/2≤k≤n‖ω(Mk,df-f)‖p.
Using Lemma 2.2, we have (3.7)ωφ2(f,1n)ω≤CKφ2(f,1n)≤C(‖ω(Mm,df-f)‖p+1n∑1≤i≤j≤d‖ωφij2Dij2Mm,df‖p)≤C1n∑k=1n(nk)δ‖ω(Mk,df-f)‖p.
Theorem 1.2 is completed.



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