# Direct and Inverse Approximation Theorems for Baskakov Operators with the Jacobi-Type Weight

Authors: Guo Feng
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We introduce a new norm and a newK-functional Kφλ(f;t)w,λ. Using this K-functional, direct and inverse approximation theorems for the Baskakov operators with the Jacobi-type weight are obtained in this paper.


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## Body

## 1. Introduction and Main Results

Letf be a function defined on the interval [0,∞).  The operators Vn(f;x) are defined as follows:(1.1)Vn(f;x)=∑k=0∞f(kn)vn,k(x),
where (1.2)vn,k(x)=(n+k-1k)xk(1+x)-n-k,
which were introduced by  Baskakov in 1957 [1]. Becker [2] and Ditzian [3] had studied these operators and obtained direct and converse theorems. In [4, 5] Totik gave a result: if f∈CB[0,+∞),0<α<1, then ∥Vn(f;x)-f(x)∥∞=O(nα) if and only if xα(1+x)α|Δh2(f;x)|≤kh2α, where h>0 and k is a positive constant. We may formulate the following question: do the Baskakov operators have similar property in the case of weighted approximation with the Jacobi weights?  It is well known that the weighted approximation is not a simple extension, because the Baskakov operators are unbounded for the usual weighted norm ∥f∥w=∥wf∥∞.  Xun and  Zhou [6]  introduced the norm (1.3)‖f‖w=‖wf‖∞+|f(0)|,f∈CB[0,∞)
and have discussed the rate of convergence for the Baskakov operators with the Jacobi weights and obtained(1.4)w(x)|Vn(f;x)-f(x)|=O(n-α)⟺K(f;t)w=O(tα),
where w(x)=xa(1+x)-b,0<a<1,b>0,0<α<1, and CB[0,∞) is the set of bounded continuous functions on [0,∞).In this paper, we introduce a new norm and a newK-functional, using the K-functional, and we get direct and inverse approximation theorems for the Baskakov operators with the Jacobi-type weight.First, we introduce some useful definitions and notations.Definition 1.1.
LetCB[0,∞) denote the set of bounded continuous functions on the interval [0,∞), and let
(1.5)Ca,b,λ={f∣f∈CB[0,∞),φ2(2-λ)wf∈CB[0,∞)},Ca,b,λ0={f∣f∈Ca,b,λ,f(0)=0},
where φ(x)=x(1+x),w(x)=xa(1+x)-b,x∈[0,∞),0≤a<λ≤1, and b≥0.
Moreover, theK-functional is given by
(1.6)Kφλ(f;t)w,λ=inf⁡g∈D{‖φ2(1-λ)(f-g)‖w+t‖φ2(2-λ)g′′‖w},
where D={g∣g∈Ca,b,λ0,g′∈A.C.loc[0,∞),∥φ2(2-λ)g′′∥w<∞}.
We are now in a position to state our main results.Theorem 1.2.
Iff∈Ca,b,λ0,  then
(1.7)‖φ2(1-λ)(Vn(f)-f)‖w≤MKφλ(f;n-1)w,λ.Theorem 1.3.
Supposef∈Ca,b,λ0,0<α<1. Then the following statements are equivalent:
(1.8)(1)φ2(1-λ)(x)w(x)|(Vn(f(x))-f(x))|=O(n-α),n≥2;(2)Kφλ(f;t)w,λ=O(tα),0<t<1.Throughout this paper,M denotes a positive constant independent of x,n, and f which may be different in different places. It is worth mentioning that for λ=1, we recover the results of [6].


## 2. Auxiliary Lemmas

To prove the theorems, we need some lemmas. By simple computation, we have(2.1)Vn′′(f;x)=n(n+1)∑k=0∞vn+2,k(x)(f(k+2n)-2f(k+1n)+f(kn))
or (2.2)Vn′′(f;x)=∑k=0∞vn,k(x)f(kn)(k(k-1)x2-2k(n+k)x(1+x)+(n+k)(n+k+1)(1+x)2).Lemma 2.1.
Letc≥0,d∈ℝ. Then
(2.3)∑k=1∞vn,k(x)(kn)-c(1+kn)-d≤Mx-c(1+x)-d,forx>0.Proof.
We notice [7]
(2.4)∑k=1∞vn,k(x)(nk)l≤Mx-l,forl∈N,∑k=0∞vn,k(x)(1+kn)m≤M(1+x)m,form∈Z.
For c=0,d=0, the result of (2.3) is obvious. For c>0,d≠0, there exists m∈ℤ, such that 0<-2d/m<1. Using Hölder's inequality, we have
(2.5)∑k=1∞vn,k(x)(kn)-c(1+kn)-d≤(∑k=1∞vn,k(x)(kn)-2c)1/2(∑k=1∞vn,k(x)(1+kn)-2d)1/2≤(∑k=1∞vn,k(x)(nk)[2c]+1)c/([2c]+1)(∑k=1∞vn,k(x)(1+kn)m)-d/m≤M(x-([2c]+1))c/([2c]+1)((1+x)m)-d/m≤Mx-c(1+x)-d.
For c>0,d=0 or c=0,d≠0, the proof is similar to that of (2.5). Thus, this proof is completed.Lemma 2.2.
Letf∈Ca,b,λ0,n∈ℕ. Then
(2.6)|w(x)φ2(1-λ)(x)Vn(f;x)|≤M‖φ2(1-λ)f‖w.Proof.
By Lemma2.1, we get
(2.7)|w(x)φ2(1-λ)(x)Vn(f;x)|=|w(x)φ2(1-λ)(x)∑k=1∞f(kn)vn,k(x)|≤‖φ2(1-λ)f‖ww(x)φ2(1-λ)(x)∑k=1∞vn,k(x)w-1(kn)φ2(λ-1)(kn)≤M‖φ2(1-λ)f‖w.Lemma 2.3.
Letf∈Ca,b,λ0,n∈ℕ. Then
(2.8)‖φ2(2-λ)Vn′′(f)‖w≤Mn‖φ2(1-λ)f‖w.Proof.
Forx∈Enc=[0,1/n],x≠0,(n+1)x(x+1)≤2n·2x≤4; using (2.1) and Lemma 2.1, we have
(2.9)|w(x)φ2(2-λ)(x)Vn′′(f;x)|≤w(x)φ2(1-λ)(x)n(n+1)x(1+x)×(∑k=0∞vn+2,k(x)w-1(k+2n)φ-2(1-λ)(k+2n)+2∑k=0∞vn+2,k(x)w-1(k+1n)φ-2(1-λ)(k+1n)+∑k=1∞vn+2,k(x)w-1(kn)φ-2(1-λ)(kn))‖φ2(1-λ)f‖w≤Mnw(x)φ2(1-λ)(x)w-1(x)φ-2(1-λ)(x)‖φ2(1-λ)f‖w≤Mn‖φ2(1-λ)f‖w.
For x∈En=(1/n,∞), by (2.2),  we get
(2.10)|w(x)φ2(2-λ)(x)Vn′′(f;x)|=|n2w(x)φ-2λ(x)∑k=1∞vn,k(x)f(kn)((kn-x)2-1+2xn(kn-x)-x(1+x)n)|≤n2w(x)φ-2λ(x)‖φ2(1-λ)f‖w∑k=1∞vn,k(x)w-1(kn)φ-2(1-λ)(kn)⋅((kn-x)2+1+2xn|kn-x|+x(1+x)n)∶=n2w(x)φ-2λ(x)‖φ2(1-λ)f‖w(I1(n,x)+I2(n,x)+I3(n,x)).
Note that for x∈En, one has the following inequality [7]
(2.11)n2mVn((t-x)2m;x)≤Mnm(φ(x))2m,m∈N.
Applying Hölder’s inequality and Lemma 2.1, we have
(2.12)I1(n,x)=∑k=1∞vn,k(x)w-1(kn)φ-2(1-λ)(kn)(kn-x)2≤(∑k=1∞vn,k(x)w-2(kn)φ-4(1-λ)(kn))1/2(∑k=1∞vn,k(x)(kn-x)4)1/2≤Mx-a-1+λ(1+x)b+λ-1x(1+x)n≤Mn-1w-1(x)φ2λ(x),(2.13)I2(n,x)=∑k=1∞vn,k(x)w-1(kn)φ-2(1-λ)(kn)|kn-x|1+2xn≤1+2xn(∑k=1∞vn,k(x)w-2(kn)φ-4(1-λ)(kn))1/2(∑k=1∞vn,k(x)(kn-x)2)1/2≤Mw-1(x)φ2λ(x)n-3/2(1+1x)1/2.
Note that for x>1/n, one has 1+1/x<2n. Hence,
(2.14)I2(n,x)≤Mn-1w-1(x)φ2λ(x),(2.15)I3(n,x)=∑k=1∞vn,k(x)w-1(kn)φ-2(1-λ)(kn)x(1+x)n≤Mn-1x(1+x)x-a-1+λ(1+x)b-1+λ=Mn-1w-1(x)φ2λ(x).
Combining (2.9)–(2.14), we get
(2.16)|w(x)φ2(2-λ)(x)Vn′′(f;x)|w≤Mn‖φ2(1-λ)f‖w.
Thus,
(2.17)‖φ2(2-λ)Vn′′(f)‖w≤Mn‖φ2(1-λ)f‖w.
The proof is completed.Lemma 2.4.
Letf∈D,n∈ℕ, and n≥2. Then
(2.18)‖φ2(2-λ)Vn′′(f)‖w≤M‖φ2(2-λ)f′′‖w.Proof.
(1)  For the caseλ≠1 or a≠0, if λ-2+b≥0, using (2.1) and Lemma 2.1, we have
(2.19)|w(x)φ2(2-λ)(x)Vn′′(f;x)|=|w(x)φ2(2-λ)(x)n(n+1)∑k=0∞vn+2,k(x)∫01/n∫01/nf′′(kn+u+v)dudv|≤‖φ2(2-λ)f′′‖w|w(x)φ2(2-λ)(x)n(n+1)∑k=0∞vn+2,k(x)⋅∫01/n∫01/n(kn+u+v)λ-a-2(1+kn+u+v)λ+b-2dudv|≤‖φ2(2-λ)f′′‖w|w(x)φ2(2-λ)(x)n(n+1)∑k=1∞vn+2,k(x)∫01/n∫01/n(kn)λ-a-2(1+k+2n)λ+b-2dudv|+‖φ2(2-λ)f′′‖w|w(x)φ2(2-λ)(x)n(n+1)vn+2,0(x)∫01/n∫01/n(u+v)λ-a-2(1+u+v)λ+b-2dudv|≤‖φ2(2-λ)f′′‖ww(x)φ2(2-λ)(x)n(n+1)∑k=1∞vn+2,k(x)n-2(kn)λ-a-2(1+k+2n)λ+b-2+3λ+b-2‖φ2(2-λ)f′′‖ww(x)φ2(2-λ)(x)n(n+1)vn+2,0(x)∫01/n11+a-λuλ-a-1du≤2⋅3λ+b-2‖φ2(2-λ)f′′‖ww(x)φ2(2-λ)(x)∑k=1∞vn+2,k(x)(kn)λ-a-2(1+kn)λ+b-2+2⋅3λ+b-2‖φ2(2-λ)f′′‖ww(x)φ2(2-λ)(x)n(n+1)vn+2,0(x)1(1+a-λ)(λ-a)(1n)λ-a≤2⋅3λ+b-2‖φ2(2-λ)f′′‖w(1+x2+a-λn(n+1)(1+a-λ)(λ-a)(1+x)n(1n)λ-a).
(i)
Ifx∈Enc,(2.20)x2+a-λn(n+1)(1+a-λ)(λ-a)(1+x)n(1n)λ-a≤n(n+1)(1+a-λ)(λ-a)(1n)2≤2(1+a-λ)(λ-a).(ii)
Ifx∈En,n≥2,(2.21)x2+a-λn(n+1)(1+a-λ)(λ-a)(1+x)n(1n)λ-a≤nλ-ax2n(n+1)(1+a-λ)(λ-a)(1+x)n(1n)λ-a≤x2n(n+1)(1+a-λ)(λ-a)n(n-1)x2≤3(1+a-λ)(λ-a).
Combining (2.19)–(2.21), we have
(2.22)|w(x)φ2(2-λ)(x)Vn′′(f;x)|≤M‖φ2(2-λ)f′′‖w.
Thus,
(2.23)‖φ2(2-λ)(x)Vn′′(f)‖w≤M‖φ2(2-λ)f′′‖w.
If λ-2+b<0, we have
(2.24)|w(x)φ2(2-λ)(x)Vn′′(f;x)|≤‖φ2(2-λ)f′′‖w|w(x)φ2(2-λ)(x)n(n+1)∑k=1∞vn+2,k(x)∫01/n∫01/n(kn)λ-a-2(1+kn)λ+b-2dudv|+‖φ2(2-λ)f′′‖w|w(x)φ2(2-λ)(x)n(n+1)vn+2,0(x)∫01/n∫01/n(u+v)λ-a-2dudv|.
By using the method similar to that of (2.19)–(2.23), it is not difficult to obtain the same inequality as (2.23).
(2) For the caseλ=1,a=0, the proof is similar to that of case (1) and even simpler. Therefore the proof is completed.Lemma 2.5 (see [8, page 200]).
LetΩ(t) be an increasing positive function on (0,a), the inequality (r>α)(2.25)Ω(h)≤M[tα+(ht)rΩ(t)]
holds true for h,t∈(0,a). Then one has
(2.26)Ω(t)=O(tα).


## 3. Proofs of Theorems

### 3.1. Proof of Theorem1.2

Proof.
First, we prove it as follows.(i)
Ifx∈Enc, then(3.1)w(x)φ2(1-λ)(x)∑k=0∞vn,k(x)|∫k/nx|kn-u|w-1(u)φ-2(2-λ)(u)du|≤Mn-1.(ii)
Ifx∈En, then(3.2)Vn((t-x)2(1+t)b-2+λ;x)≤Mn-1φ2(x)(1+x)b-2+λ.
The Proof of (3.1)
In fact, (i) fork=0, since x∈Enc, we have
(3.3)w(x)φ2(1-λ)(x)vn,0(x)∫0xuw-1(u)φ-2(2-λ)(u)du=w(x)φ2(1-λ)(x)(1+x)-n∫0xuλ-a-1(1+u)b-2+λdu.
If b-2+λ≤0, we get
(3.4)w(x)φ2(1-λ)(x)(1+x)-n∫0xuλ-a-1(1+u)b-2+λdu≤Mw(x)φ2(1-λ)(x)(1+x)-nxλ-a≤Mn-1.
If b-2+λ>0, we have
(3.5)w(x)φ2(1-λ)(x)(1+x)-n∫0xuλ-a-1(1+u)b-2+λdu≤Mw(x)φ2(1-λ)(x)(1+x)-n+b-2+λ∫0xuλ-a-1du≤Mn-1.
(ii) If k≥1, since x∈Enc, we have
(3.6)w(x)φ2(1-λ)(x)∑k=1∞vn,k(x)|∫k/nx|kn-u|w-1(u)φ-2(2-λ)(u)du|≤Mw(x)φ2(1-λ)(x)∑k=1∞vn,k(x)(kn-x)φ-2(2-λ)(x)(1+kn)b|∫k/nxu-adu|≤Mw(x)φ-2(x)∑k=1∞vn,k(x)(kn-x)(1+kn)b((kn)1-a-x1-a)≤Mw(x)φ-2(x)∑k=1∞vn,k(x)(kn-x)2-a(1+kn)b≤Mn-1.
Combining (3.4), (3.5) and (3.6), we obtain (3.1).The proof of (3.2)
Ifb-2+λ≤0, by (9.5.10) and (9.6.3) of [7], using the Cauchy-Schwarz inequality and the Hölder inequality, we obtain
(3.7)Vn((t-x)2(1+t)b-2+λ;x)≤(Vn((t-x)4;x))1/2(Vn((1+t)2(b-2+λ);x))1/2≤(Vn((t-x)4;x))1/2(Vn((1+t)-2;x))(2-b-λ)/2≤Mn-1φ2(x)(1+x)b-2+λ.
Ifb-2+λ>0, by (2.3), we get Vn((1+t)b-2+λ;x)≤M(1+x)b-2+λ, and using the Cauchy-Schwarz inequality and the Hölder inequality, we have
(3.8)Vn((t-x)2(1+t)b-2+λ;x)≤(Vn((t-x)4;x))1/2(Vn((1+t)2(b-2+λ);x))1/2≤Mn-1φ2(x)(1+x)b-2+λ.
Combining (3.7) and (3.8), we obtain (3.2).
Next, we prove Theorem1.2. For g∈D, if x∈Enc, by (3.1), we have
(3.9)|w(x)φ2(1-λ)(x)(Vn(g;x)-g(x))|=|w(x)φ2(1-λ)(x)Vn(∫xt(t-u)g′′(u)du;x)|≤w(x)φ2(1-λ)(x)‖φ2(2-λ)g′′‖wVn(|∫xt|t-u|w-1(u)φ-2(2-λ)(u)du|;x)≤M‖φ2(2-λ)g′′‖ww(x)φ2(1-λ)(x)∑k=0∞vn,k(x)|∫k/nx|kn-u|w-1(u)φ-2(2-λ)(u)du|≤Mn-1‖φ2(2-λ)g′′‖w.
If x∈En, by (3.2), we get
(3.10)|w(x)φ2(1-λ)(x)(Vn(g;x)-g(x))|=|w(x)φ2(1-λ)(x)Vn(∫xt(t-u)g′′(u)du;x)|≤M‖φ2(2-λ)g′′‖ww(x)φ2(1-λ)(x)Vn(|∫xt|t-u|w-1(u)φ-2(2-λ)(u)du|;x)≤M‖φ2(2-λ)g′′‖w|φ-2(x)Vn((t-x)2;x)+x-2-a+λw(x)φ2(1-λ)(x)Vn((t-x)2(1+t)b-2+λ;x)|≤Mn-1‖φ2(2-λ)g′′‖w.
Therefore, for f∈Ca,b,λ0,g∈D, by Lemma 2.2 and (3.9), (3.10), and the definition of Kφλ(f;n-1)w,λ, we obtain
(3.11)|w(x)φ2(1-λ)(x)(Vn(f;x)-f(x))|≤|w(x)φ2(1-λ)(x)(Vn(f-g;x))|+|w(x)φ2(1-λ)(x)(f(x)-g(x))|+|w(x)φ2(1-λ)(x)(Vn(g;x)-g(x))|≤M‖φ2(1-λ)(f-g)‖w+|w(x)φ2(1-λ)(x)(f(x)-g(x))|+|w(x)φ2(1-λ)(x)(Vn(g;x)-g(x))|≤M{‖φ2(1-λ)(f-g)‖w+n-1‖φ2(2-λ)g′′‖w}.
Taking the infimum on the right-hand side over all g∈D, we get
(3.12)|w(x)φ2(1-λ)(x)(Vn(f;x)-f(x))|≤MKφλ(f;n-1)w,λ.
This completes the proof of Theorem 1.2.


### 3.2. Proof of Theorem1.3

Proof.
By Theorem1.2, we know (2)⇒(1). Now,we will prove (1)⇒(2). In view of (1), we get
(3.13)‖φ2(1-λ)(Vn(f)-f)‖w≤Mn-α.
By the definition of K-functional, we may choose g∈D to satisfy
(3.14)‖φ2(1-λ)(f-g)‖w+n-1‖φ2(2-λ)g′′‖w≤2Kφλ(f;n-1)w,λ.
Using Lemma 2.2 and Lemma 2.3, we have
(3.15)Kφλ(f;t)w,λ≤‖φ2(1-λ)(Vn(f)-f)‖w+t‖φ2(2-λ)Vn′′(f)‖w≤Mn-α+t(‖φ2(2-λ)Vn′′(f-g)‖w+‖φ2(2-λ)Vn′′(g)‖w)≤Mn-α+t(nM‖φ2(1-λ)(f-g)‖w+M‖φ2(2-λ)g′′‖w)≤Mn-α+tnM(‖φ2(1-λ)(f-g)‖w+n-1‖φ2(2-λ)g′′‖w).
Taking the infimum on the right-hand side over all g∈D, we get
(3.16)Kφλ(f;t)w,λ≤M(n-α+tn-1Kφλ(f;n-1)w,λ).
By Lemma 2.4, we get
(3.17)Kφλ(f;n-1)w,λ≤M(n-α).
Leting (n+1)-1<t≤n-1, we get
(3.18)Kφλ(f;t)w,λ≤MKφλ(f;n-1)w,λ≤M(nn+1)-α(n+1)-α≤M(n+1)-α≤Mtα.
This completes the proof of Theorem 1.3.


## 3.1. Proof of Theorem1.2

Proof.
First, we prove it as follows.(i)
Ifx∈Enc, then(3.1)w(x)φ2(1-λ)(x)∑k=0∞vn,k(x)|∫k/nx|kn-u|w-1(u)φ-2(2-λ)(u)du|≤Mn-1.(ii)
Ifx∈En, then(3.2)Vn((t-x)2(1+t)b-2+λ;x)≤Mn-1φ2(x)(1+x)b-2+λ.
The Proof of (3.1)
In fact, (i) fork=0, since x∈Enc, we have
(3.3)w(x)φ2(1-λ)(x)vn,0(x)∫0xuw-1(u)φ-2(2-λ)(u)du=w(x)φ2(1-λ)(x)(1+x)-n∫0xuλ-a-1(1+u)b-2+λdu.
If b-2+λ≤0, we get
(3.4)w(x)φ2(1-λ)(x)(1+x)-n∫0xuλ-a-1(1+u)b-2+λdu≤Mw(x)φ2(1-λ)(x)(1+x)-nxλ-a≤Mn-1.
If b-2+λ>0, we have
(3.5)w(x)φ2(1-λ)(x)(1+x)-n∫0xuλ-a-1(1+u)b-2+λdu≤Mw(x)φ2(1-λ)(x)(1+x)-n+b-2+λ∫0xuλ-a-1du≤Mn-1.
(ii) If k≥1, since x∈Enc, we have
(3.6)w(x)φ2(1-λ)(x)∑k=1∞vn,k(x)|∫k/nx|kn-u|w-1(u)φ-2(2-λ)(u)du|≤Mw(x)φ2(1-λ)(x)∑k=1∞vn,k(x)(kn-x)φ-2(2-λ)(x)(1+kn)b|∫k/nxu-adu|≤Mw(x)φ-2(x)∑k=1∞vn,k(x)(kn-x)(1+kn)b((kn)1-a-x1-a)≤Mw(x)φ-2(x)∑k=1∞vn,k(x)(kn-x)2-a(1+kn)b≤Mn-1.
Combining (3.4), (3.5) and (3.6), we obtain (3.1).The proof of (3.2)
Ifb-2+λ≤0, by (9.5.10) and (9.6.3) of [7], using the Cauchy-Schwarz inequality and the Hölder inequality, we obtain
(3.7)Vn((t-x)2(1+t)b-2+λ;x)≤(Vn((t-x)4;x))1/2(Vn((1+t)2(b-2+λ);x))1/2≤(Vn((t-x)4;x))1/2(Vn((1+t)-2;x))(2-b-λ)/2≤Mn-1φ2(x)(1+x)b-2+λ.
Ifb-2+λ>0, by (2.3), we get Vn((1+t)b-2+λ;x)≤M(1+x)b-2+λ, and using the Cauchy-Schwarz inequality and the Hölder inequality, we have
(3.8)Vn((t-x)2(1+t)b-2+λ;x)≤(Vn((t-x)4;x))1/2(Vn((1+t)2(b-2+λ);x))1/2≤Mn-1φ2(x)(1+x)b-2+λ.
Combining (3.7) and (3.8), we obtain (3.2).
Next, we prove Theorem1.2. For g∈D, if x∈Enc, by (3.1), we have
(3.9)|w(x)φ2(1-λ)(x)(Vn(g;x)-g(x))|=|w(x)φ2(1-λ)(x)Vn(∫xt(t-u)g′′(u)du;x)|≤w(x)φ2(1-λ)(x)‖φ2(2-λ)g′′‖wVn(|∫xt|t-u|w-1(u)φ-2(2-λ)(u)du|;x)≤M‖φ2(2-λ)g′′‖ww(x)φ2(1-λ)(x)∑k=0∞vn,k(x)|∫k/nx|kn-u|w-1(u)φ-2(2-λ)(u)du|≤Mn-1‖φ2(2-λ)g′′‖w.
If x∈En, by (3.2), we get
(3.10)|w(x)φ2(1-λ)(x)(Vn(g;x)-g(x))|=|w(x)φ2(1-λ)(x)Vn(∫xt(t-u)g′′(u)du;x)|≤M‖φ2(2-λ)g′′‖ww(x)φ2(1-λ)(x)Vn(|∫xt|t-u|w-1(u)φ-2(2-λ)(u)du|;x)≤M‖φ2(2-λ)g′′‖w|φ-2(x)Vn((t-x)2;x)+x-2-a+λw(x)φ2(1-λ)(x)Vn((t-x)2(1+t)b-2+λ;x)|≤Mn-1‖φ2(2-λ)g′′‖w.
Therefore, for f∈Ca,b,λ0,g∈D, by Lemma 2.2 and (3.9), (3.10), and the definition of Kφλ(f;n-1)w,λ, we obtain
(3.11)|w(x)φ2(1-λ)(x)(Vn(f;x)-f(x))|≤|w(x)φ2(1-λ)(x)(Vn(f-g;x))|+|w(x)φ2(1-λ)(x)(f(x)-g(x))|+|w(x)φ2(1-λ)(x)(Vn(g;x)-g(x))|≤M‖φ2(1-λ)(f-g)‖w+|w(x)φ2(1-λ)(x)(f(x)-g(x))|+|w(x)φ2(1-λ)(x)(Vn(g;x)-g(x))|≤M{‖φ2(1-λ)(f-g)‖w+n-1‖φ2(2-λ)g′′‖w}.
Taking the infimum on the right-hand side over all g∈D, we get
(3.12)|w(x)φ2(1-λ)(x)(Vn(f;x)-f(x))|≤MKφλ(f;n-1)w,λ.
This completes the proof of Theorem 1.2.


## 3.2. Proof of Theorem1.3

Proof.
By Theorem1.2, we know (2)⇒(1). Now,we will prove (1)⇒(2). In view of (1), we get
(3.13)‖φ2(1-λ)(Vn(f)-f)‖w≤Mn-α.
By the definition of K-functional, we may choose g∈D to satisfy
(3.14)‖φ2(1-λ)(f-g)‖w+n-1‖φ2(2-λ)g′′‖w≤2Kφλ(f;n-1)w,λ.
Using Lemma 2.2 and Lemma 2.3, we have
(3.15)Kφλ(f;t)w,λ≤‖φ2(1-λ)(Vn(f)-f)‖w+t‖φ2(2-λ)Vn′′(f)‖w≤Mn-α+t(‖φ2(2-λ)Vn′′(f-g)‖w+‖φ2(2-λ)Vn′′(g)‖w)≤Mn-α+t(nM‖φ2(1-λ)(f-g)‖w+M‖φ2(2-λ)g′′‖w)≤Mn-α+tnM(‖φ2(1-λ)(f-g)‖w+n-1‖φ2(2-λ)g′′‖w).
Taking the infimum on the right-hand side over all g∈D, we get
(3.16)Kφλ(f;t)w,λ≤M(n-α+tn-1Kφλ(f;n-1)w,λ).
By Lemma 2.4, we get
(3.17)Kφλ(f;n-1)w,λ≤M(n-α).
Leting (n+1)-1<t≤n-1, we get
(3.18)Kφλ(f;t)w,λ≤MKφλ(f;n-1)w,λ≤M(nn+1)-α(n+1)-α≤M(n+1)-α≤Mtα.
This completes the proof of Theorem 1.3.



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*Source: 101852-2011-11-03.xml*