Meiling Wang,Dansheng Yu,?and Dejun Zhao

1Department of Mathematics,Hangzhou Normal University,Hangzhou 310036,China

2College of Fundamental Studies,Shanghai University of Engineering Science,Shanghai 201620,China

1 Introduction

Let

be the classical Jacobi weights.Let

Set

WhenI=[0,1],we briefly writekfkp,winstead ofkfkp,w,[0,1].Obviously,kfkp,wis the norm ofLpwspaces.

For anyf∈Lp([0,1]),1≤p≤∞,the corresponding Bernstein-Durrmeyer operatorsMn(f,x)are defined as follows:

where

The approximation properties ofMn(f,x)inwere also studied by Zhang(see[9]).Some approximation results were given under the restrictions

on the weight parameters.Generally speaking,the restrictions can not be eliminated for the approximation byMn(f,x).For the weighted approximation by Kantorovich-Bernstein operators defined by

the situation is similar(see[5]).Recently,Della Vecchia,Mastroianni and Szabados(see[2])introduced a weighted generalization of theKn(f,x)as follows:

Whenα=β=0,K#n(f,x)reduces to the classical Kantorovich-Bernstein operatorKn(f,x).Della Vecchia,Mastroianni and Szabados obtained the direct and converse theorems and a Voronovskaya-type relation in[2],and solved the saturation problem of the operator in[3].Their results showed thatf,x)allows a wider class of functions than the operatorKn(f,x).In fact,they dropped the restrictionsα,β<1?on the weight parameters.Later,Yu(see[8])introduced another kind of modified Bernstein-Kantorvich operators,and established direct and converse results on weighted approximation which also have no restrictionsα,β<1?

Then,a natural question is:can we modified the Bernstein-Durrmeyer operators properly such that the restrictionsα,β<1?onweighted approximation can be dropped?In the present paper,we will show that the weighted Bernstein-Durrmeyer operator introduced by Berens and Xu(see[1])is the one we need.The weighted Bernstein-Durrmeyer operator is defined as follows:

where

Define

where?(x)=pandAC(I)is the set of all absolutely continuous functions onI.Forf∈define the weighted modulus of smoothness by

with

Define

to be the best approximation offin weightedspaces by constants.

The main results of the present paper are the following:

Theorem 1.1.If f∈Lpw,1≤p≤∞,then

Theorem 1.2.If f∈Lpw,1≤p≤∞,then

2 Auxiliary lemmas

We need the following inequalities:

where

It should be noted that(2.1d)is contained in the first inequality of[2,pp.9].

Lemma 2.1.For1≤p<∞,0≤k≤n and n≥3,we have

where k?is defined by(2.2).

Proof.By the fact that(see[3])

we deduce that

Thus,we complete the proof.

Especially,by takingp=1 in(2.3),we get

Lemma 2.2.For any f∈,1≤p≤∞,we have

Proof.Whenp=∞,by(2.1a),(2.4)and(2.1b),we get

When 1

By a similar and more simpler deduction,we see that(2.6)also holds forp=1.

Combining(2.5)and(2.6),Lemma 2.2 is proved.

Lemma 2.3.If f∈,then

Proof.Direct calculations yield that(see[4,pp.331-332]),

Therefore,

Forp=∞,by(2.1a),(2.1b)and(2.1c),we have

For 1

We finish Lemma 2.3 by combining(2.8)and(2.9).

Lemma 2.4.If f∈,then

Proof.We prove the result by estimating the integral on two intervalsandEnrespectively.

Simple calculation leads to

ForI1(n,x),whenp=∞,by applying(2.1a)-(2.1c),(2.4)and Cauchy’s inequality,we have

When 1≤p<∞,by using Hlder’s inequality twice forp>1(p=1 is more direct),(2.1c),(2.1d),(2.4)and(2.1a),

ForI2(n,x),by Lemma 2.2,we have

ForI3(n,x),whenp=∞,by(2.1a)-(2.1d)and(2.4),we have

where in the last inequality,we used the fact 1/?(x)≤C,x∈En.

When 1≤p<∞,by using Hlder’s inequality,(2.1a),(2.1c),(2.1d),(2.4),and the fact 1/?(x)≤C,x∈Enagain,we deduce that

By combining(2.10)-(2.15),we already have

Now,we estimate the integral onBy(2.7),we have

When 1≤p<∞,noting thatn?2(x)≤Cforx∈Ecn,by Hlder’s inequality,(2.3)and(2.4),we have

Whenp=∞,forx∈Ecn,by(2.1a)and(2.1b),

By(2.18)and(2.19),we see that

By(2.17)and(2.20),we complete the proof of Lemma 2.4.

Lemma 2.5.For any nonnegative integer m,set

Then

and

where pi,m,n,α,β(x)are polynomials in x of fixed degree with coefficients that are bounded uniformly for all n.

Proof.Analogue to[4],we have the recursion relation:

Direct calculations yield that

and

where

By(2.23)-(2.25)and a simple induction process,we obtain(2.21)and(2.22).

By(2.21),we have

Lemma 2.6.For any given m,it holds that

Lemma 2.7.For1≤p<∞,f∈W2,pw,there is a positive constant C such that

where

Proof.Firstly,we consider the casep=1.Setg(v)=w(v)?2(v)f′(v).By the inequality(see[5]):

we have

Set

Forl≥1,by(2.1b),(2.4),and(2.26)withw≡1,we deduce that

Forl=0,by(2.1b)and(2.4),

Therefore,

Noting thatwe have

Since(see[5])

then(by takingh=(l+1))

Therefore,

Thus,we can conclude that

Now,we begin to prove the following

Forl≥1,by(2.4),and(2.26),we deduce that Forl=0,by(2.1b),(2.3)and(2.4),we have

Then,we can derive(2.30)in a similar way to the proof of(2.29).

By combining(2.28)-(2.30),we obtain Lemma 2.7 forp=1.

Finally,we prove Lemma 2.7 for 1

Set

The following maximal function inequality are well known

Since 1/w(v)≤C(1/w(t)+1/w(x))for anyvbetweenxandt,by the maximal function inequality,we have

Therefore,we only need to prove that

where kKk∞,Enis the usual supremum norm ofKonEn,and

For the first part ofK,by(2.26),

For the second part ofK,by(2.4),(2.1b)and(2.26)(withw=1),

By(2.33)and(2.34),we get(2.31),and thus Lemma 2.7 is valid for 1

3 Proofs of theorems

Proofof Theorem 1.1.It is sufficient to prove that

for?2f′′∈.By the Taylor’s formula

we have

Then,by(2.25)and(2.27),we get(3.1)immediately. ?

Proofof Theorem 1.2.The”?=”part follows from Theorem 1.1.The”=?”part can be done by using the argument of proof of Theorem 9.3.2 in[5],we omit the details here.?

References

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