# Estimation of Approximation with Jacobi Weights by Multivariate Baskakov Operator

Authors: Jianjun Wang; Haifeng Guo; Jia Jing
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We first give the unboundedness of multivariate Baskakov operators with the normal weighted norm. By introducing new norms, using the multivariate decomposition technique and the modulus of smoothness with Jacobi weight, the upper bound estimation of multivariate Baskakov operators is obtained. The obtained results not only generalize the corresponding ones for multivariate Baskakov operators without weights, but also give the approximation accuracy with the Jacobi weights approximation.


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## Body

## 1. Introduction and Main Results

LetCB[0,+∞) be the set of bounded continuous functions on [0,+∞); then the unvariate Baskakov operator is defined by
(1)Vn,1(f,x)=∑k=0∞‍vn,k(x)f(kn),
where
(2)vn,k(x)=(n+k-1k)xk(1+x)-n-k,x∈[0,∞].There are many papers to study the univariate Baskakov-type operator (see [1–4]). In [5], the authors discussed the convergence property of Baskakov operator's approximation with weights by introducing weighted smoothness modulus and obtained the characteristics of approximation. And in [6], using the weighted smoothness modulus, the close connection between the derivative of Baskakov operator and the smoothness of function approximated which has been investigated, the upper bound estimation has been established with the Jacobi weight; that is, for any f∈Cω,0≤λ≤1,
(3)|ω(x)(Vn,1(f,x)-f(x))|≤Cωφλ2(f;n-1/2φ1-λ(x))ω,
where Cω={f∣f∈C[0,∞],ωf∈L∞[0,+∞]}. In this paper, the letter C, 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.LetT∈Rd(d∈N), let T=Td=x={(x1,x2,·,xd),  and let 0≤xi<∞,0≤i≤d},
(4)x=(x1,x2⋯xd,),k=(k1,k2,…,kd),xk=x1k1x2k2⋯xdkd,|k|=k1+k2+⋯+kd|x|=x1+x2+⋯+xd,(nk)=n!k!(n-|k|),∑k=0∞‍=∑k1=0∞‍∑k2=0∞‍⋯∑kd=0∞‍,Cφr(T)={f∈CB(T):f∈CBr(T),andφirDirf∈CB(T),1≤i≤d},
where CB(T) represents the set of bounded continuous functions in T. The norm defined on T is ∥f∥∞=maxx∈T|f(x)|.Using the previous notations, the multivariate Baskakov operator is defined by(5)Vn,d(f;x)=∑k=0∞‍vn,k(x)f(kn),
where vn,k(x)=(kn+|k|-1)xk(1+x)-n-|k|. In [7], the authors gave the equivalent relation between K-functional and modulus of smoothness and obtained the following result: for any f∈CB(T),
(6)∥Vn,df-f∥∞≤Cωφ2(f,1n)∞,
where
(7)ωφr(f,t)∞=sup0≤h≤t∑i=1d‍∥Δhφieirf∥∞,Δherf(x)={0,else,∑i=0r(ri)(-1)if(x+ihe),x,x+rhe∈T.
The rth K-functional is defined as follows
(8)Kφr(f;t)=infg{∥f-g∥∞+tr∑i=1d‍(∥φirDirg∥∞),g∈Cφ2(T)}.
And the rth modulus of smoothness is
(9)ωφr(f,t)∞=sup0≤h≤t∑i=1d‍∥Δhφieirf∥∞.Naturally, we will consider the following problem: “are there similar results in the approximation with weights by multivariate Baskakov operator?” However, approximation with Jacobi weights (ω(x)=xα(1+|x|)-b(0<|α|<1,b>0)) is not a simple generalization of normal approximation means. In the norm ∥ωf∥∞, the multivariate Baskakov operator is not bounded. By introducing the weighted norm
(10)∥f∥ω=∥ωf∥∞+f(0),
we find that the multivariate Baskakov operator is bounded, and thus we can investigate approximation capability of the operators. In the previous weighted norm, the rth weighted K-functional is defined as follows:
(11)Kφr(f;t)ω=infg{∥f-g∥w+tr∑i=1d‍(∥φirDirg∥ω),g∈Cφ2(T)},f∈CB(T),
and rth weighted Ditzian-Totik modulus of smoothness is
(12)ωφr(f,t)ω=sup0≤h≤t∑i=1d‍∥Δhφieirf∥ω,g∈CB(T),
where φi2(x)=xi(1+|x|).Using the previous notations, we will present our results. Firstly, we study the unboundedness of multivariate Baskakov operator in the normal norm.Theorem 1.
Letf∈CB(T); then the operator Vn,d(f;x) is unbounded in the norm ∥ωf∥∞.In addition, introducing the new norm, we establish the upper bound estimation of weighted approximation by multivariate Baskakov operator.Theorem 2.
Letf∈CB(T); then one has
(13)∥Vn,df-f∥ω≤Cωφ2(f,1n)ω.Remark 3.
Our result reveals two things: (i) for any multivariate bounded continuous functionf∈CB(T), there is a multivariate polynomial Vn,d(f) that approximates f arbitrarily well (when n is sufficiently large) in the continuous space. (ii) Quantitatively, the approximation accuracy of a polynomial Vn,d(f) can be controlled by the ωφ2(f,1/n)ω; here ωφ2(f,1/n)ω is the weighted smoothness modulus of the function f.


## 2. Some Lemmas

In order to prove our results, we will show some lemmas in this section.Lemma 4 (see [6]).
Forf∈Cω,0≤λ≤1, one has
(14)∥Vn,1f-f∥ω≤Cωφ2(f,1n)ω.Lemma 5 (see [5]).
Forc,d≥0,
(15)|∑k=1∞‍(kn)-c(1+kn)dvn,k(x)|≤Cx-c(1+x)d,x>0.Lemma 6.
Forf∈CB(T), then one has
(16)|ω(x)∑k=1∞‍vn,k(x)f(kn)|≤C∥ωf∥∞.Proof.
For convenience, we only prove the case ofd=2; for a general d, we can prove similarly. For d=2, we have
(17)|ω(x)∑k=1∞‍vn,k(x)f(kn)|=|ω(x)∑k1=1∞‍vn,k1(x1)∑k2=1∞‍vn+k1,k2(x21+x1)f(k1n,k2n)|=|ω(x)∑k1=1∞‍vn,k1(x1)∑k2=1∞‍vn+k1,k2(x21+x1)f(k1n,k2n)×ω(k1n,k2n)(k1n)-α1(k2n)-α2×(1+k1n+k2n)-b∑k2=1∞‍vn+k1,k2(x21+x1)|≤C∥ωf∥∞|ω(x){∑k1=1∞‍vn,k1(x1)(k1n)-α1}×{∑k2=1∞‍vn+k1,k2(x21+x1)(k2n)-α2×(1+k1n+k2n)-b∑k2=1∞‍}{∑k1=1∞‍vn,k1(x1)(k1n)-α1}|=C∥ωf∥∞|ω(x){∑k1=1∞‍vn,k1(x1)(k1n)-α1}×{∑k2=1∞‍vn+k1,k2(x21+x1)(1+k2n+k1)b×(k2n+k1)-α2∑k2=1∞‍}{∑k1=1∞‍vn,k1(x1)(k1n)-α1}|≤C∥ωf∥∞∣ω(x)x1-α1x2-α2(1+x1+x2)b≤C∥ωf∥∞.
In the previous derivation, Lemma 5 and the following inequality were used:
(18)∑k=0∞‍vn,k(x)(nk+1)l≤Cx-l,l∈N.


## 3. The Proof of Theorems

In this section, we will show the unboundedness of multivariate Baskakov operator in the general weighted norm and then give the proof of Theorem2.Proof of Theorem1.
Let(19)fm(x1,x2)=1x1α1x2α2+(1/m).
Then we have
(20)∥ωfm∥∞=maxx>0|x1-α1x2-α2(1+x1+x2)-b1x1α1x2α2+(1/m)|≤(11+x1+x2)b≤1.
On the other hand,
(21)∥ωVn,df∥∞=maxx>0|ω(x)∑k=0∞‍Vn,k(x)f(kn)|≥maxx>0|ω(x)fm(0,0)vn,0(x)|-maxx>0|ω(x)∑k=1∞‍vn,k(x)f(kn)|≥maxx>0|ω(x)fm(0,0)(1+x1+x2)-n|-C≥m(32)n+b-C⟶∞(m⟶∞).As we know, multivariate Baskakov operatorVn,df has the property of preserve linear, for convenience the following discussion, we may suppose f(0)=0.Proof of Theorem2.
In order to prove Theorem2, we first show the following inequality:
(22)∥Vn,d(f)-f∥ω≤{∥f∥ω,f∈CB(T),n-1∑i=0d∥φi2Di2f∥ω,f∈Cφ2(T).
For the first inequality, it is obviously correct since∥Vn,df∥ω≤C∥f∥ω. In the following we will certify the second inequality by the mathematical induction. Our proof is based on an induction argument for the dimension d. We will also use a method called decomposition technique for the multivariate Baskakov operator.(i)
Ford=1, from Lemma 4, we have
(23)∥Vn,d(f)-f∥ω≤n-1∥φ2f′′∥ω.(ii)
Whend=r, suppose that
(24)∥Vn,d(f)-f∥ω≤∥φi2Di2f∥ωis established. So when d=r+1, using the multivariate decomposition technique in [7], we have
(25)Vn,d(f,x)=∑k1=0∞‍vn,k1(x)∑k*=0∞‍vn+k1,k*(x*1+x1),
where
(26)x*=(x2,x3,…,xn),x=(x1,x*),k*=(k2,k3,…,kd),k=(k1,k*).
Let
(27)gk1(u)=f(k1n,(1+k1n)u),u∈Td-1,h(t)=h(t,x)=f(t,(1+t)x*1+x1).
We get
(28)ω(x)(Vn,d(f,x))-f(x)=ω(x)(∑k1=0∞‍vn,k1(x1)(Vn+k1,d-1(gk1(·),z)-gk1(z)))+(Vn,1(h(·),x1)-h(x1))=P+Q,
where z=x*/(1+x1), and we have
(29)|P|=|ω(x)(∑k1=0∞vn,k1(x1)‍(Vn+k1,d-1(gk1(·),z)-gk1(z)))|=|x1α1(1+x1)-b+|α*|∑k1=0∞‍vn,k1(x1)ω(z)×(Vn+k1,d-1(gk1(·),z)-gk1(z))∑k1=0∞‍|≤x1α1(1+x1)-b+|α*|∑k1=0∞‍1n+k1vn,k1(x1)∑i=1r‍∥φi2Di2f∥ω.
The previous derivation used the following inequality:
(30)ω(x)=xα(1+|x|)-b=x1α1(x*1+x1)α*(1+x1)-b+|α*|(1+x*1+x1)-b=x1α1(1+x1)-b+|α*|ω(z),φi2(u)Di2(u)gk1(u)ω(u)=ui(1+|u|)(1+k1n)Di+12f(k1n,(1+k1n)u)×uα*(1+|u|)-b=(nk1)α1(nn+k1)-b+|α*|ωφi+12Di+12f(k1n,(1+k1n)u).
So we get
(31)∥φi2Di2gk1∥ω≤(nk1)α1(nn+k1)-b+|α*|∥φi2Di2f∥ω.
From Lemma 6, we have
(32)|P|≤|x1α1(1+x1)-b+|α*|∑k1=0∞‍1n+k1pn,k1(x1)|×(nk1)α1(nn+k1)-b+|α*|∥φi2Di2f∥ω≤Cn∥φi2Di2f∥ω.
In the following, we will establish the estimation of Q:
(33)|Q|=|(Vn,1(h(·),x1)-h(x1))ω(x)|=|x1α1(x*)α*(1+|x|)-b(Bn,1(h(·),x1)-h(x1))|=|(x*)α*(1+|x|1+x1)-b(1+x1)-bx1α1×(Vn,1(h(·),x1)-h(x1))(1+|x|1+x1)-b|≤Cn|(x*)α*(1+|x|1+x1)-b|∥φ2h′′∥ω,
and for the term ∥φ2h′′∥ω, we have the following estimation:
(34)∥φ2h′′∥ω=max0≤t<∞|t(1+t)tα1(1+t)-bh′′(t,(1+t)x*1+x1)|=max0≤t<∞|t(1+t)tα1(1+t)-b|D12f+∑i=2d‍xi1+x1D1i2f+∑i=2d‍xi1+x1Di12f+∑i=2d‍∑j=2d‍xixj(1+x1)2Di,j2f(t,(1+t)x*1+x1)=max0≤t<∞|t(1+t)tα1(1+t)-bL1|,
where
(35)L1=|1+x11+|x|φ12D12f+∑i=2d‍φ1i2D1i2f+∑i=2d‍φi12Di12f+∑i=2d‍t1+txi1+|x|φi2Di2f+∑i,j=2,i≠jd‍φij2Dij2f(t,(1+t)x*1+x1)|.
Note that
(36)ω(t,(1+t)x*1+x1)=(x*)α*(1+t1+x1)|α*|tα1(1+t)-b(1+|x|1+x1)-b.
We have
(37)∥φ2h′′∥ω=(x*)-α*(1+|x|1+x1)b(1+t1+x1)-|α*|L2.
Here
(38)|L2|=|1+x11+|x|ωφ12D12f+∑i=2d‍ωφ1i2D1i2f+∑i=2d‍ωφi12Di12f+∑i=2d‍t1+txi1+|x|ωφi2Di2f+∑i,j=2,i≠jd‍ωφij2Dij2f(t,(1+t)x*1+x1)|.
From [7], we have
(39)|Di,j2f(x)|≤sup1≤i≤d|Di2f(x)|,φi,j(x)≤min{φi(x),φj(x)},(40)∥φ2h′′∥ω=(x*)-α*(1+|x|1+x1)b(1+t1+x1)-|α*|∥φi2Di2f∥ω.
Hence
(41)|Q|≤Cn∥φi2Di2f∥ω.
Thus we have
(42)∥Vn,df-f∥ω=∥Vn,df-Vn,dg+Vn,dg-g+g-f∥ω≤C∥Vn,df-Vn,dg∥ω+∥f-g∥ω≤Cn∥φi2Di2f∥ω+∥f-g∥ω≤CKr2(f,1n)ω≤Cωφ2(f,1n)ω.
Here we used the following facts: for f∈CB(T), then
(43)1Cωφr(f,t)ω≤Kφr(f;t)ω≤Cωφr(f,t)ω.
The result is easily obtained from [7–9]. Theorem 2 is completed.



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*Source: 939565-2013-08-21.xml*