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volume 7, issue 4, article 125, 2006.

Received 06 January, 2006;

accepted 16 August, 2006.

Communicated by:N.E. Cho

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Journal of Inequalities in Pure and Applied Mathematics

ON SIMULTANEOUS APPROXIMATION FOR CERTAIN BASKAKOV DURRMEYER TYPE OPERATORS

VIJAY GUPTA, MUHAMMAD ASLAM NOOR AND MAN SINGH BENIWAL

School of Applied Sciences

Netaji Subhas Institute of Technology Sector 3 Dwarka

New Delhi 110075, India EMail:vijay@nsit.ac.in Mathematics Department

COMSATS Institute of Information Technology Islamabad, Pakistan

EMail:noormaslam@hotmail.com Department of Applied Science

Maharaja Surajmal Institute of Technology C-4, Janakpuri, New Delhi - 110058, India EMail:man_s_2005@yahoo.co.in Department of Mathematics Ch Charan Singh University Meerut 250004, India

EMail:mkgupta2002@hotmail.com

2000c Victoria University ISSN (electronic): 1443-5756 009-06

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On Simultaneous Approximation For Certain Baskakov Durrmeyer Type

Operators

Vijay Gupta, Muhammad Aslam Noor,

Man Singh Beniwal and M. K. Gupta

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Abstract

In the present paper, we study a certain integral modification of the well known Baskakov operators with the weight function of Beta basis function. We estab- lish pointwise convergence, an asymptotic formula an error estimation and an inverse result in simultaneous approximation for these new operators.

2000 Mathematics Subject Classification:41A30, 41A36.

Key words: Baskakov operators, Simultaneous approximation, Asymptotic formula, Pointwise convergence, Error estimation, Inverse theorem.

The work carried out when the second author visited Department of Mathematics and Statistics, Auburn University, USA in fall 2005.

The authors are thankful to the referee for making many valuable suggestions, lead- ing to the better presentation of the paper.

1. Introduction

For

f ∈Cγ[0,∞)≡ {f ∈C[0,∞) :|f(t)| ≤M t^γ

for some M > 0, γ > 0} we consider a certain type of Baskakov-Durrmeyer operator as

B_n(f(t), x) =

∞

X

k=1

p_n,k(x) Z ∞

0

b_n,k(t)f(t)dt+ (1 +x)⁻ⁿf(0) (1.1)

= Z ∞

0

W_n(x, t)f(t)dt where

p_n,k(x) =

n+k−1 k

x^k (1 +x)^n+k, b_n,k(t) = 1

B(n+ 1, k)· t^k−1 (1 +t)^n+k+1 and

W_n(x, t) =

∞

X

k=1

p_n,k(x)b_n,k(t) + (1 +x)⁻ⁿδ(t),

δ(t)being the Dirac delta function. The norm- || · ||_γ on the classC_γ[0,∞)is defined as||f||_γ = sup

0≤t<∞

|f(t)|t^−γ.

The operators defined by (1.1) are the integral modification of the well known Baskakov operators with weight functions of some Beta basis functions. Very

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recently Finta [2] also studied some other approximation properties of these operators. The behavior of these operators is very similar to the operators recently introduced in [6], [9] and also studied in [8]. These operators reproduce not only the constant functions but also the linear functions, which is the interesting property of such operators. The other usual Durrmeyer type integral modification of the Baskakov operators [5] reproduce only the constant functions, so one can not apply the iterative combinations easily to improve the order of approximation for the usual Baskakov Durrmeyer operators. For recent work in this area we refer to [7]. In the present paper we study some direct results which include pointwise convergence, asymptotic formula, error estimation and inverse theorem in the simultaneous approximation for the unbounded functions of growth of ordert^γ.

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2. Basic Results

In this section we mention certain lemmas which will be used in the sequel.

Lemma 2.1 ([3]). Form∈N ∪ {0}, if them^thorder moment be defined as U_n,m(x) =

∞

X

k=0

p_n,k(x) k

n −x m

, thenU_n,0(x) = 1, U_n,1(x) = 0and

nU_n,m+1(x) = x(1 +x)(U_n,m⁽¹⁾ (x) +mUn,m−1(x)).

Consequently we haveU_n,m(x) =O n^−[(m+1)/2]

.

Lemma 2.2. Let the functionT_n,m(x), m∈N ∪ {0}, be defined as T_n,m(x) =B_n (t−x)^mx

=

∞

X

k=1

pn,k(x) Z ∞

0

bn,k(t)(t−x)^mdt+ (1 +x)⁻ⁿ(−x)^m. ThenT_n,0(x) = 1, T_n,1 = 0, T_n,2(x) = ^2x(1+x)_n−1 and also there holds the recur- rence relation

(n−m)T_n,m+1(x) =x(1 +x)

T_n,m⁽¹⁾(x) + 2mTn,m−1(x)

+m(1 + 2x)T_n,m(x).

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Proof. By definition, we have

T_n,m⁽¹⁾(x) =

∞

X

k=1

p⁽¹⁾_n,k(x) Z ∞

0

b_n,k(t)(t−x)^mdt

−m

∞

X

k=1

p_n,k(x) Z ∞

0

b_n,k(t)(t−x)^m−1dt

−n(1 +x)⁻ⁿ⁻¹(−x)^m−m(1 +x)⁻ⁿ(−x)^m−1. Using the identities

x(1 +x)p⁽¹⁾_n,k(x) = (k−nx)p_n,k(x) and

t(1 +t)b⁽¹⁾_n,k(t) = [(k−1)−(n+ 2)t]b_n,k(t), we have

x(1 +x)

T_n,m⁽¹⁾(x) +mTn,m−1(x)

=

∞

X

k=1

p_n,k(x) Z ∞

0

(k−nx)b_n,k(t)(t−x)^mdt+n(1 +x)⁻ⁿ(−x)^m+1

=

∞

X

k=1

p_n,k(x) Z ∞

0

(k−1)−(n+ 2)t+ (n+ 2)(t−x) + (1 + 2x)

b_n,k(t)(t−x)^mdt+n(1 +x)⁻ⁿ(−x)^m+1

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=

∞

X

k=1

p_n,k(x) Z ∞

0

t(1 +t)b⁽¹⁾_n,k(t)(t−x)^mdt

+ (n+ 2)[T_n,m+1(x)−(1 +x)⁻ⁿ(−x)^m+1]

+ (1 + 2x)[Tn,m(x)−(1 +x)⁻ⁿ(−x)^m] +n(1 +x)⁻ⁿ(−x)^m+1

=−(m+ 1)(1 + 2x)[T_n,m(x)−(1 +x)⁻ⁿ(−x)^m]

−(m+ 2)[T_n,m+1−(1 +x)⁻ⁿ(−x)^m+1]

−mx(1 +x)[Tn,m−1(x)−(1 +x)⁻ⁿ(−x)^m−1] + (n+ 2)[T_n,m+1−(1 +x)⁻ⁿ(−x)^m+1]

+ (1 + 2x)[T_n,m(x)−(1 +x)⁻ⁿ(−x)^m] +n(1 +x)⁻ⁿ(−x)^m+1. Thus, we get

(n−m)Tn,m+1(x) =x(1 +x)[T_n,m⁽¹⁾(x) + 2mTn,m−1(x)] +m(1 + 2x)Tn,m(x).

This completes the proof of recurrence relation. From the above recurrence relation, it is easily verified for allx∈[0,∞)that

T_n,m(x) = O n^−[(m+1)/2]

.

Remark 1. It is easily verified from Lemma2.1that for eachx∈(0,∞) B_n(tⁱ, x) = (n+i−1)!(n−i)!

n!(n−1)! xⁱ+i(i−1)(n+i−2)!(n−i)!

n!(n−1)! xⁱ⁻¹+O(n⁻²).

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Corollary 2.3. Let δbe a positive number. Then for everyγ > 0, x ∈ (0,∞), there exists a constantM(s, x)independent ofnand depending onsandxsuch that

Z

|t−x|>δ

W_n(x, t)t^γdt C[a,b]

≤M(s, x)n^−s, s = 1,2,3, . . .

Lemma 2.4. There exist the polynomialsQ_i,j,r(x)independent ofnandksuch that

{x(1 +x)}^rD^r

p_n,k(x)

= X

2i+j≤r i,j≥0

nⁱ(k−nx)^jQ_i,j,r(x)p_n,k(x),

whereD≡ _dx^d.

By C₀, we denote the class of continuous functions on the interval (0,∞) having a compact support and C₀^r is the class of r times continuously differentiable functions with C₀^r ⊂ C₀. The functionf is said to belong to the gen- eralized Zygmund class Liz(α,1, a, b), if there exists a constant M such that ω₂(f, δ) ≤ M δ^α, δ > 0, whereω₂(f, δ)denotes the modulus of continuity of 2nd order on the interval[a, b]. The classLiz(α,1, a, b)is more commonly de- noted byLip^∗(α, a, b).SupposeG^(r) = {g : g ∈C₀^r+2,suppg ⊂[a⁰, b⁰]where [a⁰, b⁰]⊂(a, b)}. Forrtimes continuously differentiable functionsf withsupp f ⊂[a⁰, b⁰]the Peetre’s K-functionals are defined as

Kr(ξ, f) = inf

g∈G^(r)

h

f^(r)−g^(r)

C[a⁰,b⁰]+ξ n

g^(r)

C[a⁰,b⁰]+

g^(r+2) C[a⁰,b⁰]

oi , 0< ξ ≤1.

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For0< α <2, C₀^r(α,1, a, b)denotes the set of functions for which sup

0<ξ≤1

ξ^−α/2K_r(ξ, f, a, b)< C.

Lemma 2.5. Let 0 < a⁰ < a⁰⁰ < b⁰⁰ < b⁰ < b < ∞ and f^(r) ∈ C₀ with suppf ⊂[a⁰⁰, b⁰⁰]and iff ∈C₀^r(α,1, a⁰, b⁰),we havef^(r) ∈Liz(α,1, a⁰, b⁰)i.e.

f^(r) ∈Lip^∗(α, a⁰, b⁰)whereLip^∗(α, a⁰, b⁰)denotes the Zygmund class satisfying K_r(δ, f)≤Cδ^α/2.

Proof. Letg ∈G^(r), then forf ∈C₀^r(α,1, a⁰, b⁰),we have 4²_δf^(r)(x)

≤

4²_δ(f^(r)−g^(r))(x) +

4²_δg^(r)(x)

≤

4²_δ(f^(r)−g^(r))

C[a⁰,b⁰]+δ²

g^(r+2) C[a⁰,b⁰]

≤4M₁K_r(δ², f)≤M₂δ^α.

Lemma 2.6. Iffisrtimes differentiable on[0,∞), such thatf^(r−1) =O(t^α), α >

0ast→ ∞,then forr = 1,2,3, . . . andn > α+rwe have B^(r)_n (f, x) = (n+r−1)!(n−r)!

n!(n−1)!

∞

X

k=0

p_n+r,k(x) Z ∞

0

b_n−r,k+r(t)f^(r)(t)dt.

Proof. First

B_n⁽¹⁾(f, x) =

∞

X

k=1

p⁽¹⁾_n,k(x) Z ∞

0

b_n,k(t)f(t)dt−n(1 +x)⁻ⁿ⁻¹f(0).

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Now using the identities

p⁽¹⁾_n,k(x) =n[p_n+1,k−1(x)−p_n+1,k(x)], (2.1)

b⁽¹⁾_n,k(t) = (n+ 1)[bn+1,k−1(t)−b_n+1,k(t)].

(2.2)

fork ≥1,we have B_n⁽¹⁾(f, x)

=

∞

X

k=1

n[pn+1,k−1(x)−p_n+1,k(x)]

Z ∞

0

b_n,k(t)f(t)dt−n(1 +x)⁻ⁿ⁻¹f(0)

=npn+1,0(x) Z ∞

0

bn,1(t)f(t)dt−n(1 +x)⁻ⁿ⁻¹f(0) +n

∞

X

k=1

p_n+1,k(x) Z ∞

0

[b_n,k+1(t)−b_n,k(t)]f(t)dt

=n(1 +x)⁻ⁿ⁻¹ Z ∞

0

(n+ 1)(1 +t)⁻ⁿ⁻²f(t)dt−n(1 +x)⁻ⁿ⁻¹f(0) +n

∞

X

k=1

p_n+1,k(x) Z ∞

0

−1

nb⁽¹⁾_n−1,k+1(t)f(t)dt.

Integrating by parts, we get

B_n⁽¹⁾(f, x) =n(1 +x)⁻ⁿ⁻¹f(0) +n(1 +x)⁻ⁿ⁻¹ Z ∞

0

(1 +t)⁻ⁿ⁻¹f⁽¹⁾(t)dt

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−n(1 +x)⁻ⁿ⁻¹f(0) +

∞

X

k=1

p_n+1,k(x) Z ∞

0

bn−1,k+1(t)f⁽¹⁾(t)dt

=

∞

X

k=0

pn+1,k(x) Z ∞

0

bn−1,k+1(t)f⁽¹⁾(t)dt.

Thus the result is true for r = 1. We prove the result by induction method.

Suppose that the result is true forr=i, then B_n⁽ⁱ⁾(f, x) = (n+i−1)!(n−i)!

n!(n−1)!

∞

X

k=0

p_n+i,k(x) Z ∞

0

b_n−i,k+i(t)f⁽ⁱ⁾(t)dt.

Thus using the identities (2.1) and (2.2), we have B_n⁽ⁱ⁺¹⁾(f, x)

= (n+i−1)!(n−i)!

n!(n−1)!

×

∞

X

k=1

(n+i)[pn+i+1,k−1(x)−p_n+i+1,k(x)]

Z ∞

0

bn−i,k+i(t)f⁽ⁱ⁾(t)dt + (n+i−1)!(n−i)!

n!(n−1)! (−(n+i)(1 +x)⁻ⁿ⁻ⁱ⁻¹) Z ∞

0

bn−i,i(t)f⁽ⁱ⁾(t)dt

= (n+i)!(n−i)!

n!(n−1)! pn+i+1,0(x) Z ∞

0

bn−i,i+1(t)f⁽ⁱ⁾(t)dt

− (n+i)!(n−i)!

n!(n−1)! p_n+i+1,0(x) Z ∞

0

bn−i,i(t)f⁽ⁱ⁾(t)dt

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+ (n+i)!(n−i)!

n!(n−1)!

∞

X

k=1

p_n+i+1,k(x) Z ∞

0

[bn−i,k+i+1(t)−bn−i,k+i(t)]f⁽ⁱ⁾(t)dt

= (n+i)!(n−i)!

n!(n−1)! p_n+i+1,0(x) Z ∞

0

− 1

(n−i)b⁽¹⁾_{n−i−1,i+1}(t)f⁽ⁱ⁾(t)dt + (n+i)!(n−i)!

n!(n−1)!

∞

X

k=1

p_n+i+1,k(x) Z ∞

0

− 1

(n−i)b⁽¹⁾n−i−1,k+i+1(t)f⁽ⁱ⁾(t)dt.

Integrating by parts, we obtain

B_n⁽ⁱ⁺¹⁾(f, x) = (n+i)!(n−i−1)!

n!(n−1)!

∞

X

k=0

p_n+i+1,k(x) Z ∞

0

bn−i−1,k+i+1(t)f⁽ⁱ⁺¹⁾(t)dt.

This completes the proof of the lemma.

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3. Direct Theorems

In this section we present the following results.

Theorem 3.1. Letf ∈Cγ[0,∞)andf^(r)exists at a pointx∈(0,∞).Then we have

B_n^(r)(f, x) =f^(r)(x) asn→ ∞.

Proof. By Taylor expansion off, we have f(t) =

r

X

i=0

f⁽ⁱ⁾(x)

i! (t−x)ⁱ+ε(t, x)(t−x)^r, whereε(t, x)→0ast →x. Hence

B_n^(r)(f, x) = Z ∞

0

W_n^(r)(t, x)f(t)dt

=

r

X

i=0

f⁽ⁱ⁾(x) i!

Z ∞

0

W_n^(r)(t, x)(t−x)ⁱdt +

Z ∞

0

W_n^(r)(t, x)ε(t, x)(t−x)^rdt

=:R₁ +R₂.

First to estimateR₁, using the binomial expansion of(t−x)ⁱand Remark1, we

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have

R1 =

r

X

i=0

f⁽ⁱ⁾(x) i!

i

X

v=0

i v

(−x)^i−v ∂^r

∂x^r Z ∞

0

Wn(t, x)t^vdt

= f^(r)(x) r!

d^r dx^r

(n+r−1)!(n−r)!

n!(n−1)! x^r+terms containing lower powers ofx

=f^(r)(x)

(n+r−1)!(n−r)!

n!(n−1)!

→f^(r)(x) asn→ ∞.Next applying Lemma2.4, we obtain

R2 = Z ∞

0

W_n^(r)(t, x)ε(t, x)(t−x)^rdt,

|R₂| ≤ X

2i+j≤r i,j≥0

nⁱ |Q_i,j,r(x)|

{x(1 +x)}^r

∞

X

k=1

|k−nx|^jp_n,k(x) Z ∞

0

b_n,k(t)|ε(t, x)||t−x|^rdt+(n+r+ 1)!

(n−1)! (1 +x)^−n−r|ε(0, x)|x^r. The second term in the above expression tends to zero as n → ∞. Since ε(t, x) → 0ast → xfor a given ε > 0there exists aδsuch that|ε(t, x)| < ε whenever0 <|t−x| < δ.Ifα ≥max{γ, r}, where αis any integer, then we can find a constant M₃ > 0, |ε(t, x)(t−x)^r| ≤ M₃|t−x|^α,for |t −x| ≥ δ.

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Therefore

|R₂| ≤M₃ X

2i+j≤r i,j≥0

nⁱ

∞

X

k=0

p_n,k(x)|k−nx|^j

×

ε Z

|t−x|<δ

b_n,k(t)|t−x|^rdt+ Z

|t−x|≥δ

b_n,k(t)|t−x|^αdt

=:R₃+R₄.

Applying the Cauchy-Schwarz inequality for integration and summation respec- tively, we obtain

R₃ ≤εM₃ X

2i+j≤r i,j≥0

nⁱ ( _∞

X

k=1

p_n,k(x)(k−nx)^2j )¹₂

× ( _∞

X

k=1

pn,k(x) Z ∞

0

bn,k(t)(t−x)^2rdt )¹₂

.

Using Lemma2.1and Lemma2.2, we get

R₃ =ε·O(n^r/2)O(n^−r/2) =ε·o(1).

Again using the Cauchy-Schwarz inequality, Lemma2.1and Corollary2.3, we

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get

R₄ ≤M₄ X

2i+j≤r i,j≥0

nⁱ

∞

X

k=1

p_n,k(x)|k−nx|^j Z

|t−x|≥δ

b_n,k(t)|t−x|^αdt

≤M₄ X

2i+j≤r i,j≥0

nⁱ

∞

X

k=1

|t−x|≥δ

b_n,k(t)dt ¹₂

× Z

|t−x|≥δ

bn,k(t)(t−x)^2αdt ¹₂

≤M₄ X

2i+j≤r i,j≥0

nⁱ ( _∞

X

k=1

p_n,k(x)(k−nx)^2j )¹₂

× ( _∞

X

k=1

p_n,k(x) Z ∞

0

b_n,k(t)(t−x)^2αdt )¹₂

= X

2i+j≤r i,j≥0

nⁱO(n^j/2)O(n^−α/2) =O(n^(r−α)/2) =o(1).

Collecting the estimates ofR₁−R₄,we obtain the required result.

Theorem 3.2. Letf ∈C_γ[0,∞).Iff^(r+2)exists at a pointx∈(0,∞).Then

n→∞lim n

B_n^(r)(f, x)−f^(r)(x)

=r(r−1)f^(r)(x) +r(1 + 2x)f^(r+1)(x) +x(1 +x)f^(r+2)(x).

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Proof. Using Taylor’s expansion off,we have f(t) =

r+2

X

i=0

f⁽ⁱ⁾(x)

i! (t−x)ⁱ +ε(t, x)(t−x)^r+2,

where ε(t, x) → 0as t → x and ε(t, x) = O((t−x)^γ), t → ∞ for γ > 0.

Applying Lemma2.2, we have n

B_n^(r)(f(t), x)−f^(r)(x)

=n

"_r+2 X

i=0

f⁽ⁱ⁾(x) i!

Z ∞

0

W_n^(r)(t, x)(t−x)ⁱdt−f^(r)(x)

#

+n Z ∞

0

W_n^(r)(t, x)ε(t, x)(t−x)^r+2dt

=:E₁+E₂. First, we have

E₁ =n

r+2

X

i=0

f⁽ⁱ⁾(x) i!

i

X

j=0

i j

(−x)^i−j Z ∞

0

W_n^(r)(t, x)t^jdt−nf^(r)(x)

= f^(r)(x) r! n

B_n^(r)(t^r, x)−(r!) + f^(r+1)(x)

(r+ 1)! n

(r+ 1)(−x)B_n^(r)(t^r, x) +B_n^(r)(t^r+1, x) + f^(r+2)(x)

(r+ 2)! n

"

(r+ 2)(r+ 1)

2 x²B_n^(r)(t^r, x)

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+ (r+ 2)(−x)B_n^(r)(t^r+1, x) +B_n^(r)(t^r+2, x)

# . Therefore, by Remark1, we have

E₁ =nf^(r)(x)

(n+r−1)!(n−r)!

n!(n−1)! −1

+ nf^(r+1)(x) (r+ 1)!

(x−1)(−x)

(n+r−1)!(n−r)!

n!(n−1)!

+

(n+r)!(n−r−1)!

n!(n−1)! (r+ 1)!x+ (r+ 1)r(n+r−1)!(n−r−1)!

n!(n−1)! r!

+ nf^(r+2)(x) (r+ 2)!

(r+ 2)(r+ 1)

2 x²(n+r−1)!(n−r)!

n!(n−1)! r!

+ (r+ 2)(−x)

(n+r)!(n−r−1)!

2 x(r+ 1)!

+(r+ 1)r(n−r−1)!(n−r−1)!

n!(n−1)! r!

+

(n+r+ 1)!(n−r−2)!

n!(n−1)!

(r+ 2)!

2 x² + (r+ 2)(r+ 1) (n+r)!(n−r−2)!

n!(n−1)! (r+ 1)!x

+O(n⁻²).

In order to complete the proof of the theorem it is sufficient to show thatE₂ →0 as n → ∞ which easily follows proceeding along the lines of the proof of Theorem3.1and by using Lemma2.1, Lemma2.2and Lemma2.4.

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Lemma 3.3. Let0 < α <2,0 < a < a⁰ < a⁰⁰ < b⁰⁰ < b⁰ < b < ∞.Iff ∈ C₀ withsuppf ⊂[a⁰⁰, b⁰⁰]and

Bn^(r)(f,·)−f^(r)

C[a,b]=O(n^−α/2),then K_r(ξ, f) =M₅

n^−α/2+nξK_r(n⁻¹, f) . ConsequentlyK_r(ξ, f)≤M₆ξ^α/2, M₆ >0.

Proof. It is sufficient to prove

K_r(ξ, f) =M₇{n^−α/2+nξK_r(n⁻¹, f)},

for sufficiently large n. Because suppf ⊂ [a⁰⁰, b⁰⁰] therefore by Theorem 3.2 there exists a functionh⁽ⁱ⁾ ∈G^(r), i=r, r+ 2,such that

B_n^(r)(f,•)−h⁽ⁱ⁾

C[a⁰,b⁰]≤M8n⁻¹. Therefore,

K_r(ξ, f)≤3M₉n⁻¹+

B_n^(r)(f,•)−f^(r) _C[a₀_,b₀_]

+ξn

B_n^(r)(f,•)

C[a⁰,b⁰]+

B_n^(r+2)(f,•) C[a⁰,b⁰]

o . Next, it is sufficient to show that there exists a constantM₁₀such that for each g ∈G^(r)

(3.1)

B_n^(r+2)(f,•)

C[a⁰,b⁰]≤M10n{

f^(r)−g^(r)

C[a⁰,b⁰]+n⁻¹

g^(r+2) C[a⁰,b⁰].

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Also using the linearity property, we have (3.2)

B_n^(r+2)(f,•) C[a⁰,b⁰]

≤

B^(r+2)_n (f −g,•)

C[a⁰,b⁰]+

B_n^(r+2)(g,•) C[a⁰,b⁰]. Applying Lemma2.4, we get

Z ∞

0

∂^r+2

∂x^r+2Wn(x, t)

dt ≤ X

2i+j≤r+2 i,j≥0

∞

X

k=1

nⁱ|k−nx|^j |Q_i,j,r+2(x)|

{x(1 +x)}^r+2

×p_n,k(x) Z ∞

0

b_n,k(t)dt+ d^r+2

dx^r+2[(1 +x)⁻ⁿ].

Therefore by the Cauchy-Schwarz inequality and Lemma2.1, we obtain

(3.3)

B_n^(r)(f −g,•)

C[a⁰,b⁰]≤M₁₁n

f^(r)−g^(r) C[a⁰,b⁰],

where the constantM11is independent off andg.Next by Taylor’s expansion, we have

g(t) =

r+1

X

i=0

g⁽ⁱ⁾(x)

i! (t−x)ⁱ+ g^(r+2)(ξ)

(r+ 2)! (t−x)^r+2,

where ξ lies between t and x. Using the above expansion and the fact that R∞

0

∂^m

∂x^mW_n(x, t)(t−x)ⁱdt = 0form > i,we get (3.4)

B_n^(r+2)(g,•) C[a⁰,b⁰]

≤M₁₂

g^(r+2)

C[a⁰,b⁰]·

Z ∞

0

∂^r+2

∂x^r+2W_n(x, t)(t−x)^r+2dt C[a⁰,b⁰]

.

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Also by Lemma2.4and the Cauchy-Schwarz inequality, we have E ≡

Z ∞

0

∂^r+2

∂x^r+2W_n(x, t)

(t−x)^r+2dt

≤ X

2i+j≤r+2 i,j≥0

∞

X

k=1

nⁱp_n,k(x)|k−nx|^j |Q_i,j,r+2(x)|

{x(1 +x)}^r+2 Z ∞

0

b_n,k(t)(t−x)^r+2dt

+ d^r+2

dx^r+2[(−x)^r+2(1 +x)⁻ⁿ]

≤ X

2i+j≤r+2 i,j≥0

|Qi,j,r+2(x)|

{x(1 +x)}^r+2

∞

X

k=1

pn,k(x)(k−nx)^2j

!¹₂

×

∞

X

k=1

p_n,k(x) Z ∞

0

b_n,k(t)(t−x)^2r+4dt

!¹₂ Z ∞

0

b_n,k(t)dt ¹₂

+ d^r+2

dx^r+2[(−x)^r+2(1 +x)⁻ⁿ]

= X

2i+j≤r+2 i,j≥0

nⁱ |Q_i,j,r+2(x)|

{x(1 +x)}^r+2O(n^j/2)O

n⁻(¹⁺^r2) .

Hence

(3.5) ||B_n^(r+2)(g,•)||_C[a⁰_,b⁰_] ≤M₁₃||g^(r+2)||_C[a⁰_,b⁰_].

Combining the estimates of (3.2)-(3.5), we get (3.1). The other consequence follows form [1]. This completes the proof of the lemma.

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Theorem 3.4. Let f ∈ C_γ[0,∞) and suppose 0 < a < a₁ < b₁ < b < ∞.

Then for allnsufficiently large, we have B_n^(r)(f,•)−f^(r)

_C[a

1,b1]≤max

M₁₄ω₂

f^(r), n⁻¹², a, b

+M₁₅n⁻¹kfk_γ , whereM₁₄=M₁₄(r), M₁₅ =M₁₅(r, f).

Proof. For sufficiently small δ >0, we define a functionf_2,δ(t)corresponding tof ∈Cγ[0,∞)by

f_2,δ(t) = δ⁻² Z ^δ₂

−^δ

2

Z ₂^δ

−^δ

2

f(t)−∆²_ηf(t) dt₁dt₂,

where η = ^t¹^+t₂ ², t ∈ [a, b]and ∆²_ηf(t)is the second forward difference of f with step lengthη. Following [4] it is easily checked that:

(i) f_2,δ has continuous derivatives up to order2kon[a, b], (ii) kf_2,δ^(r)k_C[a₁_,b₁_]≤Mc₁δ^−rω₂(f, δ, a, b),

(iii) kf−f_2,δk_C[a₁_,b₁_]≤Mc₂ω₂(f, δ, a, b), (iv) kf_2,δk_C[a₁_,b₁_]≤Mc₃kfk_γ,

whereMc_i, i= 1,2,3are certain constants that depend on[a, b]but are independent off andn[4].

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We can write B_n^(r)(f,•)−f^(r)

C[a1,b1]

≤

B_n^(r)(f −f_2,δ,•)

C[a1,b1]+

B_n^(r)(f_2,δ,•)−f_2,δ^(r) C[a1,b1]

+

f^(r)−f_2,δ^(r) C[a1,b1]

=:H1 +H2+H3. Sincef_2,δ^(r) = f^(r)

2,δ(t), by property (iii) of the functionf_2,δ,we get H₃ ≤Mc₄ω₂(f^(r), δ, a, b).

Next on an application of Theorem3.2, it follows that

H₂ ≤Mc₅n⁻¹

r+2

X

j=r

f_2,δ^(j)

C[a,b]

.

Using the interpolation property due to Goldberg and Meir [4], for each j = r, r+ 1, r+ 2, it follows that

f_2,δ^(j)

C[a1,b1]

≤Mc₆

||f_2,δ||_C[a,b]+ f_2,δ^(r+2)

C[a,b]

.

Therefore by applying properties (iii) and (iv) of the of the function f_2,δ, we obtain

H2 ≤Mc74·n⁻¹

||f||γ+δ⁻²ω2(f^(r), δ) .

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Finally we shall estimateH₁, choosinga^∗, b^∗ satisfying the conditions0< a <

a^∗ < a1 < b1 < b^∗ < b < ∞. Supposeψ(t)denotes the characteristic function of the interval[a^∗, b^∗], then

H₁ ≤

B_n^(r)(ψ(t)(f(t)−f_2,δ(t)),•) C[a1,b1]

+

B_n^(r)((1−ψ(t))(f(t)−f_2,δ(t)),•) C[a1,b1]

=:H₄+H₅. Using Lemma2.6, it is clear that

B_n^(r) ψ(t)(f(t)−f_2,δ(t)), x

= (n+r−1)!(n−r)!

n!(n−1)!

∞

X

k=0

p_n+r,k(x) Z ∞

0

bn−r,k+r(t)ψ(t)(f^(r)(t)−f_2,δ^(r)(t))dt.

Hence

B_n^(r)(ψ(t)(f(t)−f2,δ(t)),•)

C[a1,b1]≤Mc8

f^(r)−f_2,δ^(r)

C[a^∗,b^∗]. Next for x ∈ [a₁, b₁]and t ∈ [0,∞)\[a^∗, b^∗], we choose a δ₁ > 0 satisfying

|t−x| ≥δ₁.

Therefore by Lemma2.4and the Cauchy-Schwarz inequality, we have I ≡B_n^(r)((1−ψ(t))(f(t)−f_2,δ(t), x)

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and

|I| ≤ X

2i+j≤r i,j≥0

nⁱ |Q_i,j,r(x)|

{x(1 +x)}^r

∞

X

k=1

pn,k(x)|k−nx|^j

× Z ∞

0

b_n,k(t)(1−ψ(t))|f(t)−f_2,δ(t)|dt +(n+r−1)!

(n−1)! (1 +x)^−n−r(1−ψ(0))|f(0)−f_2,δ(0)|.

For sufficiently largen, the second term tends to zero. Thus

|I| ≤Mc₉||f||_γ X

2i+j≤r i,j≥0

nⁱ

∞

X

k=1

|t−x|≥δ1

b_n,k(t)dt

≤Mc₉||f||_γδ₁^−2m X

2i+j≤r i,j≥0

nⁱ

∞

X

k=1

p_n,k(x)|k−nx|^j Z ∞

0

b_n,k(t)dt ¹₂

× Z ∞

0

b_n,k(t)(t−x)^4mdt ¹₂

≤Mc₉||f||_γδ₁^−2m X

2i+j≤r i,j≥0

nⁱ ( _∞

X

k=1

p_n,k(x)(k−nx)^2j )¹₂

× ( _∞

X

k=1

pn,k(x) Z ∞

0

bn,k(t)(t−x)^4mdt )¹₂

.

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Hence by using Lemma2.1and Lemma2.2, we have I ≤Mc₁₀||f||_γδ₁^−2mO

n⁽ⁱ⁺^j²^−m)

≤Mc₁₁n^−q||f||_γ,

where q = m − ^r₂. Now choosing m > 0 satisfying q ≥ 1, we obtain I ≤ Mc₁₁n⁻¹kfk_γ.Therefore by property (iii) of the functionf_2,δ(t),we get

H₁ ≤Mc₈

f^(r)−f_2,δ^(r)

_C[a_∗_,b_∗_]+Mc₁₁n⁻¹||f||_γ

≤Mc₁₂ω₂(f^(r), δ, a, b) +Mc₁₁n⁻¹||f|k_γ. Choosingδ=n⁻¹², the theorem follows.

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4. Inverse Theorem

This section is devoted to the following inverse theorem in simultaneous approximation:

Theorem 4.1. Let 0 < α < 2,0 < a₁ < a₂ < b₂ < b₁ < ∞ and suppose f ∈C_γ[0,∞).Then in the following statements(i)⇒(ii)

(i) ||Bn^(r)(f,•)||_C[a₁_,b₁_] =O(n^−α/2), (ii) f^(r) ∈Lip^∗(α, a2, b2),

whereLip^∗(α, a₂, b₂)denotes the Zygmund class satisfyingω₂(f, δ, a₂, b₂)≤ M δ^α.

Proof. Let us choosea⁰, a⁰⁰, b⁰, b⁰⁰in such a way thata1 < a⁰ < a⁰⁰ < a2 < b2 <

b⁰⁰ < b⁰ < b₁.Also supposeg ∈C₀^∞withsuppg ∈[a⁰⁰, b⁰⁰]andg(x) = 1on the interval[a₂, b₂].Forx∈[a⁰, b⁰]withD≡ _dx^d,we have

B_n^(r)(f g, x)−(f g)^(r)(x)

=D^r(B_n((f g)(t)−(f g)(x)), x)

=D^r(B_n(f(t)(g(t)−g(x)), x)) +D^r(B_n(g(x)(f(t)−f(x)), x))

=:J₁+J₂.

Using the Leibniz formula, we have J₁ = ∂^r

∂x^r Z ∞

0

W_n(x, t)f(t)[g(t)−g(x)]dt

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=

r

X

i=0

r i

Z ∞

0

W_n⁽ⁱ⁾(x, t) ∂^r−i

∂x^r−i[f(t)(g(t)−g(x))]dt

=−

r−1

X

i=0

r i

g^(r−i)(x)B_n⁽ⁱ⁾(f, x) + Z ∞

0

W_n^(r)(x, t)f(t)(g(t)−g(x))dt

=:J₃+J₄.

Applying Theorem3.4, we have

J₃ =−

r−1

X

i=0

r i

g^(r−i)(x)f⁽ⁱ⁾(x) +O n⁻^α² ,

uniformly in x∈ [a⁰, b⁰].Applying Theorem3.2, the Cauchy-Schwarz inequality, Taylor’s expansions off andgand Lemma2.2, we are led to

J₄ =

r

X

i=0

g⁽ⁱ⁾(x)f^(r−i)(x)

i!(r−i)! r! +o n⁻¹²

=

r

X

i=0

r i

g⁽ⁱ⁾(x)f^(r−i)(x) +o n⁻^α² , uniformly inx∈[a⁰, b⁰].Again using the Leibniz formula, we have

J2 =

r

X

i=0

r i

Z ∞

0

W_n⁽ⁱ⁾(x, t) ∂^r−i

∂x^r−i[g(t)(f(t)−f(x))]dt

=

r

X

i=0

r i

g^(r−i)(x)B_n⁽ⁱ⁾(f, x)−(f g)^(r)(x)