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volume 5, issue 3, article 79, 2004.

Received 21 January, 2004;

accepted 14 June, 2004.

Communicated by:C.P. Niculescu

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Journal of Inequalities in Pure and Applied Mathematics

RATE OF CONVERGENCE FOR A GENERAL SEQUENCE OF DURRMEYER TYPE OPERATORS

NIRAJ KUMAR

School of Applied Sciences

Netaji Subhas Institute of Technology Sector 3, Dwarka

New Delhi 110045, India.

EMail:neeraj@nsit.ac.in

c

2000Victoria University ISSN (electronic): 1443-5756 016-04

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Rate Of Convergence For A General Sequence Of Durrmeyer Type Operators

Niraj Kumar

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J. Ineq. Pure and Appl. Math. 5(3) Art. 79, 2004

http://jipam.vu.edu.au

Abstract

In the present paper we give the rate of convergence for the linear combinations of the generalized Durrmeyer type operators which includes the well known Szasz-Durrmeyer operators and Baskakov-Durrmeyer operators as special cases.

2000 Mathematics Subject Classification:41A16, 41A25.

Key words: Szasz-Durrmeyer operators, Baskakov-Durrmeyer operators, Rate of convergence, Order of approximation.

The author is extremely grateful to the referee for making valuable suggestions lead- ing to the better presentation of the paper.

1. Introduction

Durrmeyer [1] introduced the integral modification of Bernstein polynomials so as to approximate Lebesgue integrable functions on the interval [0,1].We now consider the general family of Durrmeyer type operators, which is defined by

(1.1) S_n(f(t), x) = (n−c)

∞

X

v=0

p_n,v(x) Z ∞

0

p_n,v(t)f(t)dt, x∈I

where n ∈ N, n > max{0,−c}and p_n,v(x) = (−1)^{v x}_v!^vϕ^(v)n (x). Also {φ_n} is a sequence of real functions having the following properties on[0, a], where a >0and for alln∈N, v ∈N∪ {0}, we have

(I) φ_n ∈C^∞[0, a], φ_n(0) = 1.

(II) φ_nis complete monotonic.

(III) There existc∈N:φ^(v+1)n =−nφ^(v)_n+c, n >max{0,−c}. Some special cases of the operators (1.1) are as follows:

1. Ifc= 0,φ_n(x) = e^−nx, we get the Szasz-Durrmeyer operator.

2. Ifc = 1, φ_n(x) = (1 +x)⁻ⁿ, we obtain the Baskakov-Durrmeyer operator.

3. Ifc > 1, φn(x) = (1 +cx)⁻ⁿ^c , we obtain a general Baskakov-Durrmeyer operator.

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4. Ifc=−1, φ_n(x) = (1−x)ⁿ, we obtain the Bernstein-Durrmeyer operator.

Very recently Srivastava and Gupta [9] studied a similar type of operators and obtained the rate of convergence for functions of bounded variation. It is easily verified that the operators (1.1) are linear positive operators and these operators reproduce the constant ones, while the operators studied in [9] reproduce every linear functional for the casec= 0. Several researchers studied different approximation properties on the special cases of the operators (1.1), the pioneer work on Durrmeyer type operators is due to S. Guo [3], Vijay Gupta (see e. g.

[4], [5]), R. P. Sinha et al. [8] and Wang and Guo [11], etc. It turns out that the order of approximation for such type of Durrmeyer operators is at bestO(n⁻¹), how so ever smooth the function may be. In order to improve the order of approximation, we have to slacken the positivity condition of the operators, for this we consider the linear combinations of the operators (1.1). The technique of linear combinations is described as follows:

S_n,k(f, x) =

1 d⁻¹₀ d⁻²₀ ... d^−k₀ 1 d⁻¹₁ d⁻²₁ ... d^−k₁ ... .. .. ... ..

... .. .. ... ..

1 d⁻¹_k d⁻²_k ... d^−k_k

−1

S_d₀_n(f, x) d⁻¹₀ d⁻²₀ ... d^−k₀ S_d₁_n(f, x) d⁻¹₁ d⁻²₁ ... d^−k₁

... .. .. ... ..

S_d_k_n(f, x) d⁻¹_k d⁻²_k ... d^−k_k .

Such types of linear combinations were first considered by May [7] to improve the order of approximation of exponential type operators. In the alternative form

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the above linear combinations can be defined as

S_n,k(f, x) =

k

X

j=0

c(j, k)S_d_j_n(f, x),

where

C(j, k) =

k

Y

i=0 i6=j

dj

d_j −d_i, k6= 0 and C(0,0) = 1.

Iff ∈ L_p[0,∞),1 ≤p < ∞and0< a₁ < a₃ < a₂ < b₂ < b₃ < b₁ <∞ andI_i = [a_i, b_i], i= 1,2,3, the Steklov meanf_η,mofm^thorder corresponding tof is defined as

f_η,m(t) = η^−m Z ^η₂

−η 2

· · · Z ^η₂

−η 2

f(t) + (−1)^m−1∆^m^P^m

i=1tif(t)Y^m

i=1

dt_i; t∈I₁.

It can be verified [6,10] that

(i) f_η,mhas derivative up to orderm,fη,m^(m−1) ∈AC(I₁)andfη,m^(m)exist almost every where and belongs toL_p(I₁);

(ii) fη,m^(r)

_L

p(I1) ≤K₁η^−rω_r(f, η, p, I₁),r= 1,2,3, . . . , m;

(iii) kf−f_η,mk_L

p(I2) ≤K₂ω_m(f, η, p, I₁);

(iv) f_η,m^m

Lp(I2) ≤K₃η^mkfk_L

p(I1)

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(v) fη,m^(m)

Lp(I1) ≤K4η^−rωr(f, η, p, I1)

whereK_i⁰s, i= 1,2,3,4are certain constants independent off andη.

In the present paper we establish the rate of convergence for the combinations of the generalized Durrmeyer type operators inL_p−norm.

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2. Auxiliary Results

To prove the rate of convergence we need the following lemmas:

Lemma 2.1. Let m ∈ N∪ {0}, alsoµ_n,m(x)is them^th order central moment defined by

µ_n,m ≡S_n((t−x)^m, x) = (n−c)

∞

X

v=0

p_n,v(x) Z ∞

0

p_n,v(t)(t−x)^mdt,

then

(i) µ_n,m(x)is a polynomial inxof degreem.

(ii) µ_n,m(x)is a rational function innand for each0 ≤x < ∞ µ_n,m(x) = O n^−[(m+1)/2]

.

Remark 2.1. Using Hölder’s inequality, it can be easily verified thatSn(|t−x|^r, x)

=O(n^−r/2)for eachr >0and for every fixed0≤x <∞.

Lemma 2.2. For sufficiently largenandq ∈N, there holds Sn,k((t−x)^q, x) = n^−(k+1){F(q, k, x) +o(1)},

where F(q, k, x)are certain polynomials in xof degree q and0 ≤ x < ∞is arbitrary but fixed.

Proof. For sufficiently large values ofnwe can write, from Lemma2.1 S_n((t−x)^q, x) = Q0(x)

n^[(q+1)/2] + Q1(x)

n[(q+1)/2]+1 +· · ·+Q_[q/2](x) n^q ,

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whereQ_i(x),i= 0,1,2, . . . are certain polynomials inxof at most degreeq.

ThereforeS_n,k((t−x)^q, x)is given by

1 d⁻¹₀ d⁻²₀ ... d^−k₀ 1 d⁻¹₁ d⁻²₁ ... d^−k₁ ... .. .. ... ..

... .. .. ... ..

1 d⁻¹_k d⁻²_k ... d^−k_k

−1

×

Q0(x)

(d0n)^[(q+1)/2] + _(d ^Q¹^(x)

0n)[(q+1)/2]+1 +.... d⁻¹₀ d⁻²₀ ... d^−k₀

Q0(x)

(d1n)^[(q+1)/2] + _(d ^Q¹^(x)

1n)[(q+1)/2]+1 +.... d⁻¹₁ d⁻²₁ ... d^−k₁ ... .. .. ... ..

... .. .. ... ..

Q0(x)

(d_kn)^[(q+1)/2] +_(d ^Q¹^(x)

kn)[(q+1)/2]+1 +.... d⁻¹_k d⁻²_k ... d^−k_k

=n^−(k+1){F(q, k, x) +o(1)}, 0≤x <∞.

Lemma 2.3 ([4]). Let1≤p <∞,f ∈L_p[a, b],f^(m) ∈AC[a, b]andf^(m+1)∈ Lp[a, b], then

f^(j)

Lp[a,b] ≤C n

f^(m+1)

Lp[a,b]+kfk_L

p[a,b]

o , j = 1,2,3, . . . , mandCis a constant depending onj, p, m, a, b.

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Lemma 2.4. Let f ∈ L_p[0,∞), p > 1. If f^(2k+1) ∈ AC(I₁) and f^(2k+2) ∈ L_p(I₁), then for allnsufficiently large

(2.1) kS_n,k(f,·)−fk_L

p(I2) ≤C₁n^−(k+1)n

f^(2k+2)

Lp(I2)+kfk_L

p[0,∞)

o . Also if f ∈ L1[0,∞), f^(2k+1) ∈ L1(I1) with f^(2k) ∈ AC(I1) and f^(2k+1) ∈ BV(I₁), then for allnsufficiently large

(2.2) kS_n,k(f,·)−fk_L

p(I2)

≤C₂n^−(k+1)n

f^(2k+1)

BV(I1)+

f^(2k+1)

L1(I2)+kfk_L

1[0,∞)

o . Proof. First letp >1. By the hypothesis, for allt ∈[0,∞)andx∈I₂, we have

S_n,k(f, x)−f(x) =

2k+1

X

i=1

f⁽ⁱ⁾(x)

i! S_n,k((t−x)ⁱ, x)

+ 1

(2k+ 1)!S_n,k(φ(t) Z t

x

(t−w)^2k+1f^(2k+2)(w)dw, x) +S_n,k(F(t, x)(1−φ(t)), x)

=E1+E2+E3, say,

whereφ(t)denotes the characteristic function ofI₁and

F(t, x) = f(t)−

2k+1

X

i=0

(t−x)ⁱ

i! f⁽ⁱ⁾(x).

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Applying Lemma2.2and Lemma2.3, we have kE₁k_L

p(I2) ≤C₃n^−(k+1)

2k+1

X

i=1

f⁽ⁱ⁾ Lp(I2)

≤C4n^−(k+1) n

kfk_L_p_(I

2)+

f^(2k+2) Lp(I2)

o .

Next we estimate E2. LetHf be the Hardy Littlewood maximal function [10]

off^(2k+2)onI₁, using Hölder’s inequality and Lemma2.1, we have

|E₂| ≤ 1

(2k+ 2)!S_n,k

φ(t)

Z t x

|t−w|^2k+1

f^(2k+2)(w) dw

, x

≤ 1

(2k+ 2)!S_n,k

φ(t)|t−x|^2k+1

Z t x

f^(2k+2)(w) dw

, x

≤ 1

(2k+ 2)!S_n,k

φ(t)|t−x|^2k+2|H_f(t)|, x

≤ 1

(2k+ 2)!

n S_n,k

φ(t)|t−x|^q(2k+2), xo¹_q

{S_n,k(φ(t)|H_f(t)|^p, x)}¹^p

≤C5n^−(k+1)

k

X

j=0

C(j, k) Z b1

a1

(djn−c)

∞

X

v=0

pdjn,v(x)pdjn,v(t)|Hf(t)|^pdt

!¹_p . Next applying Fubini’s theorem, we have

kE₂k^p_L

p(I2)

≤C₆n^−p(k+1)

k

X

j=0

C(j, k) Z b2

a2

Z b1

a1

(d_jn−c)

∞

X

v=0

p_d_j_n,v(x)p_d_j_n,v(t)|H_f(t)|^pdtdx

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≤C₆n^−p(k+1)

k

X

j=0

C(j, k) Z b1

a1

Z b2

a2

(d_jn−c)

∞

X

v=0

p_d_j_n,v(x)p_d_j_n,v(t)dx

!

|H_f(t)|^pdt

≤C₇n^−p(k+1)kH_fk^p_L

p(I1) ≤C₈n^−p(k+1)

f^(2k+2)

p Lp(I1). Therefore

kE2k_L

p(I2)≤C8n^−(k+1)

f^(2k+2) Lp(I1).

Fort∈[0,∞)\[a1, b1], x∈I2there exists aδ >0such that|t−x| ≥δ.Thus E₃ ≡S_n,k(F(t, x)(1−φ(t)), x))

≤δ^−(2k+2)

k

X

j=0

C(j, k)S_d_j_n |F(t, x)|(t−x)^2k+2, x

≤δ^−(2k+2)

k

X

j=0

C(j, k)

"

S_d_j_n |f(t)|(t−x)^2k+2, x

+

2k+1

X

i=0

f⁽ⁱ⁾(x) i! S_d_j_n

|t−x|^2k+2+i, x

#

=E31+E32, say.

Applying Hölder’s inequality and Lemma2.1, we have

|E₃₁| ≤δ^−(2k+2)

k

X

j=0

C(j, k)

S_d_j_n(|f(t)|^p, x)

1 p n

S_d_j_n(|t−x|^q(2k+2), x)o¹_q

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≤C₉

k

X

j=0

C(j, k)

S_d_j_n(|f(t)|^p, x)

1

p 1

(d_jn)^k+1. Finally by Fubini’s theorem, we obtain

kE₃₁k^p_L

p(I2) = Z b2

a2

|E₃₁|^pdx

≤C9 k

X

j=0

C(j, k)(djn)^−p(k+1)

× Z b2

a2

Z ∞ 0

(d_jn−c)p_d_j_n,v(x)p_d_j_n,v(t)|f(t)|^pdtdx

≤C10n^−p(k+1)kfk^p_L

p[0,∞). Again by Lemma2.3, we have

kE₃₂k_L

p(I2) ≤C₁₁n^−(k+1)n kfk_L

p(I2)+

f^(2k+2) Lp(I2)

o . Thus

kE₃k_L

p(I2) ≤C₁₂n^−(k+1)n kfk_L

p[0,∞)+

f^(2k+2) Lp(I2)

o . Combining the estimates ofE₁, E₂, E₃, we get (2.1).

Next suppose p = 1. By the hypothesis, for almost all x ∈ I₂ and for all

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t ∈[0,∞), we have S_n,k(f, x)−f(x) =

2k+1

X

i=1

f⁽ⁱ⁾(x)

i! S_n,k((t−x)ⁱ, x)

+ 1

(2k+ 1)!S_n,k(φ(t) Z t

x

(t−w)^2k+1f^(2k+1)(w)dw, x) +S_n,k(F(t, x)(1−φ(t)), x)

=M₁+M₂+M₃, say,

whereφ(t)denotes the characteristic function ofI1 and F(t, x) = f(t)−

2k+1

X

i=0

(t−x)ⁱ

i! f⁽ⁱ⁾(x), for almost allx∈I2 andt∈[0,∞).

Applying Lemma2.2and Lemma2.3, we have kM1k_L

1(I2)≤C13n^−(k+1) n

kfk_L

1(I2)+

f^(2k+1) L1(I2)

o . Next, we have

kM₂k_L

1(I2)

≤ 1

(2k+ 1)!

k

X

j=0

|C(j, k)|

Z b2

a2

Z b1

a1

(djn−c)

∞

X

v=0

pdjn,v(x)pdjn,v(t)|t−x|^2k+1

×

Z t x

df^(2k+1)(w)

dtdx.

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For eachd_jnthere exists a non negative integerr=r(d_jn)satisfying r(d_jn)^−1/2 <max(b₁−a₂, b₂−a₁)≤(r+ 1)(d_jn)^−1/2. Thus

kM2k_L

1(I2) ≤ 1 (2k+ 1)!

k

X

j=0

|C(j, k)|

r

X

l=0

Z b2

a2

(Z x+(l+1)(djn)^−1/2 x+(l)(djn)^−1/2

φ(t)(djn−c)

.

∞

X

v=0

p_d_j_n,v(x)p_d_j_n,v(t)|t−x|^2k+1

) Z x+(l+1)(djn)^−1/2 x

φ(w)

df^(2k+1)(w)

! dtdx

+ 1

(2k+ 1)!

k

X

j=0

|C(j, k)|

r

X

l=0

Z b2

a2

(Z x−(l)(d_jn)^−1/2 x+(l+1)(djn)^−1/2

φ(t)(d_jn−c)

×

∞

X

v=0

p_d_j_n,v(x)p_d_j_n,v(t)|t−x|^2k+1 )

· Z x

x−(l+1)(djn)^−1/2

φ(w)

df^(2k+1)(w)

! dtdx.

Supposeφx,c,s(w)denotes the characteristic function of the interval

x−c(djn)^−1/2, x+s(d_jn)^−1/2

, wherec, sare nonnegative integers. Then we have kM2k_L

1(I2)

≤ 1

(2k+ 1)!

k

X

j=0

|C(j, k)|

r

X

l=1

Z b2

a2

(Z x+(l+1)(djn)^−1/2

x+(l)(djn)^−1/2

φ(t)(d_jn)²(d_jn−c)

×

∞

X

v=0

pdjn,v(x)pdjn,v(t)l⁻⁴|t−x|^2k+5 )

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×

Z x+(l+1)(djn)^−1/2 x

φ(w)φ_x,0,l+1(w)

df^(2k+1)(w)

! dtdx

+ 1

(2k+ 1)!

k

X

j=0

|C(j, k)|

r

X

l=1

Z b2

a2

(Z x−(l)(d_jn)^−1/2

x+(l+1)(djn)^−1/2

φ(t)(d_jn)²(d_jn−c).

×

∞

X

v=0

b_d_j_n,v(x)p_d_j_n,v(t)l⁻⁴|t−x|^2k+5 )

× Z x

x−(l+1)(djn)^−1/2

φ(w)φ_x,l+1,0(w)

df^(2k+1)(w)

! dtdx

× 1

(2k+ 1)!

k

X

j=0

|C(j, k)|

Z b2

a2

(Z a1+(djn)^−1/2 (djn)^−1/2

φ(t)(d_jn−c)

×

∞

X

v=0

×

Z x+(djn)^−1/2 x−(djn)^−1/2

φ(w)φ_x,1,1(w)

df^(2k+1)(w)

! dtdx

≤ 1

(2k+ 1)!

k

X

j=0

|C(j, k)|

r

X

l=1

l⁻⁴ Z b2

a2

(Z x+(l+1)(djn)^−1/2 x+(l)(djn)^−1/2

φ(t)(d_jn−c)

×

∞

X

v=0

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× Z b1

a1

φ_x,0,l+1(w)

df^(2k+1)(w)

dtdx

+ 1

(2k+ 1)!

k

X

j=0

|C(j, k)|

r

X

l=1

l⁻⁴ Z b2

a2

(Z x−(l)(d_jn)^−1/2

x+(l+1)(djn)^−1/2

φ(t)(d_jn−c)

×

∞

X

v=0

× Z b1

a1

φ_x,l+1,0(w)

df^(2k+1)(w)

dtdx

+ 1

(2k+ 1)!

k

X

j=0

|C(j, k)|

Z b2

a2

(Z a1+(djn)^−1/2

(djn)^−1/2

φ(t)(d_jn−c)

×

∞

X

v=0

× Z b1

a1

φ(w)φ_x,1,1(w)

df^(2k+1)(w)

dtdx.

Applying Lemma2.1and using Fubini’s theorem we get kM₂k_L

1(I2)

≤C14 k

X

j=0

|C(j, k)|(djn)⁻^2k+1²

r

X

l=1

l⁻⁴ Z b2

a2

Z b1

a1

φx,0,l+1(w)

df^(2k+1)(w) dx

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+C₁₄

k

X

j=0

|C(j, k)|(d_jn)⁻^2k+1²

r

X

l=1

l⁻⁴ Z b2

a2

Z b1

a1

φ_x,l+1,0(w)

df^(2k+1)(w) dx +

Z b1

a1

φ_x,1,1(w)

df^(2k+1)(w) dx

≤C14 k

X

j=0

|C(j, k)|(djn)⁻^2k+1²

r

X

l=1

l⁻⁴ Z b1

a1

Z b2

a2

φx,0,l+1(w)dx

df^(2k+1)(w)

+C₁₄

k

X

j=0

|C(j, k)|(d_jn)⁻^2k+1²

r

X

l=1

l⁻⁴ Z b1

a1

Z b2

a2

φ_x,l+1,0(w)

df^(2k+1)(w) dx +

Z b2

a2

φ_x,1,1(w)

df^(2k+1)(w) dx

≤C₁₅

k

X

j=0

|C(j, k)|(d_jn)⁻^2k+1²

r

X

l=1

l⁻⁴ (Z b1

a1

Z w

w−(l+1)(djn)⁻¹²

dx

!

df^(2k+1)(w)

)

+C₁₅

k

X

j=0

|C(j, k)|(d_jn)⁻^2k+1²

r

X

l=1

l⁻⁴ Z b1

a1











Z w+(l+1)(djn)⁻¹² w

dx





df^(2k+1)(w)

+ Z b1

a1





Z w+(djn)⁻¹² w−(d_jn)⁻¹²

dx





df^(2k+1)(w)







≤C₁₆n^−(k+1)

f^(2k+1)

B.V.(I1).

Finally we estimate M₃. It is sufficient to choose the expression without the

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linear combinations. For all t ∈ [0,∞)\[a₁, b₁] and all x ∈ I₂, we choose a δ >0such that|t−x| ≥δ. Thus

kS_n(F(t, x)(1−φ(t)), x)k_L

1(I2)

≤ Z b2

a2

Z ∞ 0

(n−c)

∞

X

v=0

p_n,v(x)p_n,v(t)|f(t)|(1−φ(t))dtdx

≤

2k+1

X

i=0

1 i!

Z b2

a2

Z ∞ 0

(n−c)

∞

X

v=0

p_n,v(x)p_n,v(t)

f⁽ⁱ⁾(t)

.|t−x|ⁱ(1−φ(t))dtdx

=M₄+M₅, say.

For sufficiently largetthere exist positive constantsC₁₇, C₁₈such that^(t−x)_t2k+2^2k+2+1 >

C₁₇for allt≥C₁₈, t∈I₂. Applying Fubini’s theorem and Lemma2.1, we obtain

M₄ =

Z C18

0

Z b2

a2

+ Z ∞

C18

Z b2

a2

^∞ X

v=0

(n−c)p_n,v(x)p_n,v(t)|f(t)|(1−φ(t))dtdx

≤C₁₉n^−(k+1)

Z C20

0

|f(t)|dt

+ 1 C₁₇

Z ∞ C20

Z b2

a2

∞

X

v=0

p_n,v(x)p_n,v(t)(t−x)^2k+2

(t^2k+2+ 1)|f(t)|dxdt

≤C20n^−(k+1)

Z C20

0

|f(t)|dt

+ Z ∞

C20

|f(t)|dt

≤C₂₁n^−(k+1)kfk_L

1[0,∞).

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Finally by the Lemma2.3, we have M₅ ≤C₂₂n^−(k+1)n

kfk_L

1(I2)+

f^(2k+1) L1(I2)

o . CombiningM₄andM₅, we obtain

M₃ ≤C₂₃n^−(k+1)n kfk_L

1[0,∞)+

f^(2k+1) L1(I2)

o . This completes the proof of (2.2) of the lemma.

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3. Rate Of Convergence

Theorem 3.1. Letf ∈Lp[0,∞), p≥1. Then fornsufficiently large kSn,k(f,·)−fk_L

p(I2)≤C24

n

ω2k+2(f, n^−1/2, p, I1) +n^−(k+1)kfk_L_p_[0,∞)o , whereC₂₄is a constant independent off andn.

Proof. We can write kS_n,k(f,∗)−fk_L

p(I2)

≤ kS_n,k(f−f_η,2k+2,∗)k_L

p(I2)

+kS_n,k(f_η,2k+2,∗)−f_η,2k+2)k_L

p(I2)+k(f_η,2k+2−f)k

Lp(I2)

=E₁+E₂+E₃, say.

It is well known that f_η,2k+2^(2k+1)

B.V.(I3)

= f_η,2k+2^(2k+2)

L1(I3)

. Therefore from Lemma2.4(p >1)and(p= 1)we have

E₂ ≤C₂₅n^−(k+1)

f_η,2k+2^(2k+2)

Lp(I3)+kf_η,2k+2k_L

p[0,∞)

≤C₂₆n^−(k+1)

n^−(k+2)ω_2k+2(f, η, p, I₁) +kfk_L

p[0,∞)

, which follows from the properties of Steklov means.

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Letφ(t)be the characteristic function ofI₃, we have S_n((f−f_η,2k+2)(t), x)

=S_n(φ(t)(f −f_η,2k+2)(t), x) +S_n((1−φ(t))(f−f_η,2k+2)(t), x)

=E₄+E₅, say.

By Hölder’s inequality Z b2

a2

|E₄|^pdx≤ Z b2

a2

Z b1

a1

(n−c)

∞

X

v=0

p_n,v(x)p_n,v(t)|(f−f_η,2k+2)(t)|^pdtdx.

Applying Fubini’s theorem, we have Z b2

a2

|E4|^pdx≤ kf −fη,2k+2k_Lp(I

2). Similarly, for allp≥1

kE₅k

Lp(I2)

≤C₂₇η^−(k+1)kf −f_η,2k+2k_Lp(0,∞). Consequently, via the property of Steklov means, we find that kS_n(f −f η,2k+ 2,·)k_Lp(I2) ≤C₂₈n

ω_2k+2(f, η, p, I₁) +η^−(k+1)kfk_Lp[0,∞)o . Hence

E₁ ≤C₂₉n

ω_2k+2(f, η, p, I₁) +η^−(k+1)kfk_Lp[0,∞)o . Thus, withη=n^−1/2, the result follows.

This completes the proof of the theorem.

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References

[1] J.L. DURRMEYER, Une Formula d’ inversion de la Transformee de Laplace: Application a la Theorie des Moments, These de 3c cycle, Fac- ulte des Science de l’ University de Paris, 1967.

[2] S. GOLDBERG ANDV. MEIR, Minimum moduli of ordinary differential operators, Proc. London Math. Soc., 23 (3) (1971), 1–15.

[3] S. GUO, On the rate of convergence of Durrmeyer operator for functions of bounded variation, J. Approx. Theory, 51(1987), 183–192.

[4] V. GUPTA, Simultaneous approximation by Szasz-Durrmeyers operators, The Math. Student, 64 (1-4) (1995), 27–36.

[5] V. GUPTA, A note on modified Baskakov operators, Approx.Theory and its Appl., 10(3) (1994), 74–78.

[6] E. HEWITTANDK. STROMBERG, Real and Abstract Analysis, McGraw Hill, New York (1956).

[7] C.P. MAY, Saturation and inverse theorem for combination of a class of exponential type operators, Canad. J. Math., 28 (1976), 1224–1250.

[8] R.P. SINHA, P.N. AGRAWALANDV. GUPTA, On simultaneous approx- imation by modified Baskakov operators, Bull Soc. Math Belg. Ser. B., 44 (1991), 177–188.

[9] H.M. SRIVASTAVAANDV. GUPTA, A certain family of summation inte- gral type operators, Math and Comput. Modelling, 37 (2003), 1307–1315.

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[10] A.F. TIMAN, Theory of Approximation of Functions of a Real Variable, Hindustan Publ. Corp., Delhi 1966.

[11] Y. WANG AND S. GUO, Rate of approximation of function of bounded variation by modified Lupas operators, Bull. Austral. Math. Soc., 44 (1991), 177–188.