Approximation by Some Linear Positive Operators in Lp Spaces · 2015. 1. 25. · -analogue of classical Baskakov operators. In the recent years for similar type of operators some

International Journal of Scientific and Innovative Mathematical Research (IJSIMR)

Volume 2, Issue 3, March 2014, PP 289-300

ISSN 2347-307X (Print) & ISSN 2347-3142 (Online)

www.arcjournals.org

©ARC Page | 289

Approximation by Some Linear Positive Operators in Lp

Spaces

Rupa Sharma Research Scholar, Mewar University

Chittorgarh (Rajasthan), India

[email protected]

Prerna M. Sharma

Department of Mathematics

SRM University, NCR Campus,

Modinagar (U.P), India

[email protected]

Abstract: The summation integral type Baskakov operator was introduced by Gupta and Srivastava [4] in

1993. In the present paper, we extend our studies and introduce the Baskakov- Szasz Stancu operators. We

prove a direct theorem for the linear combinations of Baskakov- Szasz Stancu type operators. To prove our

main theorem, we use the technique of linear approximation method viz. Steklov mean.

Keywords: Steklov mean, linear combinations, linear positive operators, Stancu operators

Ams subject classification: 41A25, 41A35

1. INTRODUCTION

For ),[0, Lpf 1,p modified Baskakov - Szasz operators defined by Gupta and Srivastava

[4] in 1993 are as

),[0,,)()()(=),(,

0,

0=

xdttftqxpnxfSknkn

k

n (1)

In 1983, the Stancu type generalization of Bernstein operators was given in [9]. In 2010 in [1], the

authors have studied the Stancu type generalization of the q -analogue of classical Baskakov

operators. In the recent years for similar type of operators some approximation properties have

been discussed by Maheshwari [7], Maheshwari-Sharma [8] etc. Motivated by the recent work on

Stancu type operators, here we propose the Stancu type generalization of Baskakov-Szasz Stancu

operators, for 0 as

),[0,,)()(=),(,

0,

0=

,,

xdtn

ntftqxpnxfS

knkn

k

n

(2)

where

,)(1

1=)(

, kn

k

kn

x

x

k

knxp

.

!

)(=)(

,

k

ntetq

knt

kn

(3)

On putting 0== operators (2) reduce to operators defined in (1). It is observed that the

order of approximation for these operators is ).(1

nO To improve the order of approximation, we

consider the linear combinations of these operators ),,(,,

xkfSn

of the operators

),(,,

xfSn

jd

as

),,(),(=),,(,,

0=

,,xfSkjCxkfS

nj

d

k

j

n (4)

mailto:[email protected]

Rupa Sharma & Prerna M. Sharma

International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page |290

where

0

( , ) = , 0 (0 , 0 ) = 1 .

k

j

i j i

i j

d iC j k k a n d C

d d

(5)

We can write the operators (2) as

,),,(=),(0

,,dt

n

ntftxnWxfS

n

(6)

where

).()(=),,(,,

0=

tqxpntxnWknkn

k

The th

k linear combinations ),,,( xkfSn

considered by May [6] for the operators ),(,,

xfSn

jd

are defined by

,

...),(

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

...),(

...),(

...1

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

...1

...1

=),,(

1

1

1

11

0

1

00

1

1

1

1

1

0

1

0

k

kknk

d

k

nd

k

nd

k

kk

k

k

n

ddxfS

ddxfS

ddxfS

dd

dd

dd

xkfS

(7)

where k

dddd ,...,,,210

are 1)( k arbitrary but fixed distinct positive numbers. In this paper we

have considered <<<,0<<<<<<<0123321

babbbaaa and 1,2,3=],,[= ibaIiii

.

We denote by .H For )[0, Lpf and <1 p , the Steklov mean m

f,

of th

m order

corresponding to f is defined by

],,...,,)(1)()([.....=)(21

1/2

/2

/2

/2

/2

/2, m

m

h

mm

mdtdtdttftftf

where i

m

iut 1=

= and )( tfm

h is the

thm order forward difference of function f with step

length ,h defined as

)].()([=)(=)(111

tfhtftftfm

hh

m

h

m

h

From [10, 5] we have

(1) m

f,

has derivative up to order ),(,1

1)(

,IACfm

m

m

and

1)(

,

m

mf

exists a.e and belong to

);(1

ILp

(2) ;1,2,......=),,,,(1

)2

(

)(

,mrIpfHf

r

r

Ip

L

r

m

(3) );,,,(1

)2

(,

IpfHffm

Ip

Lm

(4) ;)

1()

2(

, Ip

LIp

Lm

fHf

(5) ,)

1()

2(

)(

, Ip

L

m

Ip

L

r

mfHf

;1,2,...,= mr

Approximation by Some Linear Positive Operators in Lp Space

International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page | 291

Here we represent absolute continuous function on ],[ ba as ],[ baAC and H are certain

constants defined on ,I but are independent of f and n . ],[ baBV denotes the set of all

functions of bounded variation on ],[ ba . The semi norm ],[ baBV

f is defined by the total

variation of f on ].,[ ba For ,<<],1,[ pbaLfp

the Hardy-Little wood majorant of f is

defined as

).(,)(1

sup=)( badttfx

xhx

x

f

In the present work we establish some direct results on Lp -norm for the linear combination of the

Baskakov- Szasz- Stancu operators.

2. MOMENTS ESTIMATION AND AUXILIARY RESULTS

In this section we estimate moments and mention certain basic results.

Lemma 1 ..[4] Let the th

m order moment be defined by

,)()(=,

0,

0=

,,,dtx

n

nttqxpnT

m

knkn

k

n

(8)

then 1,=)(,0

xTn

n

xxT

n

1=)(

,1

and 2

2

22

22

,2

)(

)2(1

)(

)(2

)(

)2(=)(

nn

nnx

n

nnxxT

n

and we have the recurrence relation 1m N

)()(1)(1)()()(1=)(1,,,1,

xTxn

xxmxTmxTxxxnTmnmnmnmn

(9)

Consequently for 0,x

,=)(1)/2][(

,

m

mnnOxT (10)

where ][ denotes the integral part of . By using Holder’s inequality we get the conclusion,

for every fixed ).[0, x

0>,=,||/2

,,,rnOxxtS

rr

n

(11)

Lemma 2. For Np and n sufficiently large there hold,

)[0,,(1)),,(=,,)(1)(

,,,

toxkpQnxkxtS

kp

n

where ),,( xkpQ are certain polynomials in x of degree /2p .

Proof of above Lemma is easy and can be seen on similar type of operators.

Lemma 3. [3] Let ,1 p ],,[ baLfp

],[)(

baACfk and ],,[

1)(baLpf

k

then

],[],[

1)(

],[

)(

bap

Lbap

L

k

bap

L

jffHf

where ,1,2,...,= kj and H are certain constants depending only on .,,,, bapkj

Lemma 4 . [2] There exist the polynomials )(,,

xqrji

on ),[0, independent of n and k such



that

, , , ,

2

, 0

(1 ) ( ) = ( ) ( ) ( ) .

r k

r r i j

n k i j r n kr

i j r

i j

dx x p x n k n x q x p x

d x

3. DIRECT ESTIMATES

Theorem 1. Let 1.>),[0, pLfp

If f has 2)(2 k derivative on 1

I with )(1

1)(2IACf

k

then for n sufficiently large

,,.),()[0,)

2(

1)(1)(

)2

(,,

pLI

pL

kk

Ip

Ln

ffHnfkfS

where the constant H is independent of n and f

Proof: By our assumptions, for 2

Ix and ,1

It we have

duufun

nt

kxf

j

xn

nt

n

ntf

k

k

t

x

j

j

k

j

)(1)!(2

1)(

!=

2)(2

12

)(

12

0=

,1,

n

ntx

n

ntF (12)

where )( t denotes the characteristic function on .1

I

).(!

)(

=,)(

12

0=

xfj

xn

nt

n

ntfx

n

ntF

j

j

k

j

For all )[0, t and .2

Ix Using (12) in (4), we have

xkxn

ntS

j

xfxfxkfS

j

n

jk

j

n,,

!

)(=)(),,(

,,

)(12

1=

,,

xkduufun

nt

n

ntS

k

k

k

t

xn

,,)(1)!(2

1 2)(2

12

,,

.=:,,1,321,,

xk

n

ntx

n

ntFS

n

According to Lemma 2 and [3]

)2

(

)(

12

1=

1)(

)2

(1I

pL

j

k

j

k

Ip

LfHn

.)

2(

2)(2

)2

(

)(1)(

Ip

L

k

Ip

L

jkffHn

For finding 2

I , let f

h be the Hardy-Littlewood Majorant [11] of 2)(2 k

f on .1

I Now using

Holder’s inequality (11), we obtain



xduufun

nt

n

ntSR

k

k

t

xn

,)(=2)(2

12

,,1

xduufun

nt

n

ntS

k

k

x

tn

,)(22

12

,,

xduufxn

nt

n

ntS

kt

x

k

n,)(

22

12

,,

xn

nthx

n

nt

n

ntS

f

k

n,

22

,,

p

p

fn

qqk

nx

n

nt

n

nthSx

n

ntx

n

ntS

1/

,,

1/2)(2

,,,,

pp

fn

kx

n

nt

n

nthSHn

1/

,,

1)(,

.),,(

1/

1

1

1)(

pp

f

b

a

kdt

n

nthtxnWHn

Using Fubini’s theorem and [12], we get

dtdxn

nthtxnWHnR

p

f

b

a

b

a

pkp

Ip

L

),,(

1

1

2

2

1)(

)2

(1

dtn

nthdxtxnWHn

p

f

b

a

b

a

pk

),,(

2

2

1

1

1)(

dtn

nthHn

p

f

b

a

pk

1

1

1)(

p

Ip

L

f

pk

n

nthHn

)1

(

1)(

.)

1(

2)(21)(p

Ip

L

kpkfHn

Hence,

.)

1(

2)(21)(

)2

(1I

pL

kpk

Ip

LfHnR

Consequently,

.)

1(

2)(21)(

)2

(2I

pL

kpk

Ip

LfHn



For ],[\)[0,11

bat , 2

Ix , 0> such that . xt

Thus

x

n

ntx

n

ntFS

n,1,

,,

xx

n

ntx

n

ntFS

k

n

k,,

22

,,

2)(2

xxn

ntxf

j

xn

nt

n

ntfS

k

j

j

k

j

n

k,)(

!

22

)(

12

0=

,,

2)(2

xx

n

nt

n

ntfS

k

n

k,[

22

,,

2)(2

.=],!

)(

32

22

,,

)(12

1=

RRxxn

ntS

j

xfjk

n

jk

j

Using Holder’s inequality and (11), we get

qqk

n

pp

n

kxx

n

ntSxxfSR

1/2)(2

,,

1/

,,

2)(2

2,,)(

.,

1/

,,

1)(

pp

n

kx

n

ntfSHn

Again applying Fubini’s theorem, we get

dtdxn

ntftxnWHndtR

p

b

a

pkpb

a

),,(

0

2

2

1)(

2

2

2

.)[0,

1)(

pL

kfHn

Thus

.)[0,

1)(

)2

(2

pL

k

Ip

LfHnR

Now using (11) and [3], we get )

2(

)(

12

0=

1)(

)2

(1I

pL

j

k

j

k

Ip

LfHnR

.)

2(

2)(2

)2

(

)(1)(

Ip

L

k

Ip

L

jkffHn

Combining the estimates of 2

R and ,1

R we get the result

.)

2(

2)(2

)[0,

)(1)(

)2

(2

Ip

L

k

pL

jk

Ip

LffHn



Which is the required theorem.

Theorem 2 . Let ).[0,1

Lf If f has 1)(2 k derivatives on 1

I with )(1

)(2IACf

k and

),(1

1)(2IBVf

k

then for n sufficiently large we have

,,.),()[0,)

2(

1

1)(2

)2

(

1)(21)(

)2

(1

,,

pLIL

k

IBV

kk

ILn

fffHnfkfS

where the constant H is independent of n and .f

Proof: By our given hypothesis on f , and for all 2

Ix and for all ,1

It we have

)).()((1)!(2

1)(

!

)(=)(

1)(212)(

12

0=

udfutk

xfi

xttf

kkt

x

i

ik

i

We can write

n

ntudfu

n

nt

kxf

i

xn

nt

n

ntf

k

k

t

x

i

i

k

i

)(1)!(2

1)(

!=

1)(2

12

)(

12

0=

.1,

n

ntx

n

ntF

where )( t being the characteristic function on .1

I

),(!

)(

=,)(

12

0=

xfi

xn

nt

n

ntfx

n

ntF

i

i

k

i

For all )[0, t and .2

Ix Therefore we have

xkxn

ntS

i

xfxfxkfS

i

n

ik

i

n,,

!

)(=)(),,(

,,

)(12

0=

,,

xkn

ntudfu

n

ntS

k

k

k

t

xn

,,)(1)!(2

1 1)(2

12

,,

.=:,,1,321,,

RRRxkn

ntx

n

ntFS

n

Applying Lemma 1 and [3], we get

.)

2(

1

1)(2

)2

(1

1)(

)2

(1

1

IL

k

IL

k

ILffHnR

Now we have,

)2

(1

1)(2

12

,,,)(=

IL

k

k

t

xn

xn

ntudfu

n

ntSG



.)(),,(1)(2

12

1

1

2

2

dtdxudfxn

nttxnW

kt

x

k

b

a

b

a

.

For each n there exists a nonnegative integer )(= nrr such that

.1)(),(max<1/2

1221

1/2 nrababrn Then we have

121/2

1)(

1/2)(

2

20=

),,({

k

nlx

nlx

b

a

r

l

xn

nttxnW

n

ntG

dtudfu

knlx

x

)()(1)(2

1/21)(

121/2

1/21)(

),,(

k

lnx

nlx

xn

nttxnW

n

nt

.})().(

1)(2

1/21)(

dxdtudfuk

x

nlx

Let )(,,

udcx

be the characteristic function of the interval ],,[1/21/2

dnxcnx where ,c d

are non-negative integers. Hence we get

52

24

1/21)(

1/2)(

2

21=

),,({

k

nlx

nlx

b

a

r

l

xn

ntnltxnW

n

ntG

dtudfuuk

lx

nlx

x

)()()(1)(2

1,0,

1/21)(

52

24

1/2

1/21)(

),,(

k

lnx

nlx

xn

ntnltxnW

n

nt

dxdtudfuuk

lx

x

nlx

})()()(1)(2

1,0,1/21)(

121/2

1

1/2

2

2

),,(

k

na

n

b

a

xn

nttxnW

n

nt

dtdxudfuuk

lx

nx

nl

)()()(1)(2

,1,

1/2

1/2

521/2

1)(

1/2)(

2

2

24

1=

),,({[

k

nlx

nlx

b

a

r

l

xn

nttxnW

n

ntnl

])()()(1)(2

1,0,

1

1

dtudfuuk

lx

b

a

521/2

)(

1/21)(

),,(

k

nlx

nlx

xn

nttxnW

n

nt

dxdtudfuk

lx

b

a

})()(1)(2

1,0,

1

1

121/2

1

1/2

2

2

),,(

k

na

n

b

a

xn

nttxnW

n

nt



.)()(1)(2

,1,

1

1

dtdxudfuk

lx

b

a

Further using Lemma 1 and Fubini’s theorem, we obtain

dxudfulHnGk

lx

b

a

b

a

r

l

k)()([[

1)(2

1,0,

1

1

2

2

4

1=

1)/2(2

])()()()(1)(2

,1,

1

1

2

2

1)(2

1,0,

1

1

2

2

dxudfudxudfuk

lx

b

a

b

a

k

lx

b

a

b

a

)()([=1)(2

1,0,

2

2

1

1

4

1=

1)/2(2udfdxulHn

k

lx

b

a

b

a

r

l

k

)()())()(1)(2

,1,

2

2

1

1

1)(2

1,0,

2

2

1

1

udfdxuudfdxuk

lx

b

a

b

a

k

lx

b

a

b

a

)([1)(2

1/21)(

2

2

4

1=

1)/2(2udfdxlHn

kw

nlw

b

a

r

l

k

)()(1)(2

1/2

1/2

1

1

1)(2

1/21)(

1

1

udfdxudfdxk

nw

nw

b

a

knlw

w

b

a

.)

1(

1)(21)(

IBV

kkfHn

Hence, ,)

1(

1)(21)(

)2

(2IBV

kk

Ip

LfHnR

where the constant H depends on .k

For ,],,[\)[0,211

Ixbat there exist a 0> such that . xt Then

dtdxn

nttftxnWx

n

ntxtFS

pb

a

Ip

L

n

1)(),,(,1),(

0

2

2

)2

(

,,

i

ib

a

k

i

xn

ntxftxnW

i

)(),,(

!

1 )(

0

2

2

12

0=

dtdxn

nt

1

.=54

RR

Now for sufficiently large ,t positive constant 0

N and .H such that ,>

1

22

22

H

n

nt

xn

nt

k

k

for all ,0

Mt .2

Ix

Now using Fubini’s theorem

.=1),,(=76

2

2

0

0

2

2

0

04

RRdtdxn

nt

n

ntftxnWR

b

aN

b

a

N

Now by using Lemma 1, we have



dtdxxn

nt

n

ntftxnWR

k

b

a

Nk

2)(2

2

2

0

0

2)(2

6),,(=

.0

0

1)(

dt

n

ntfHn

Nk

.

1

),,(1

=0

1)(

22

22

2

20

7

dtn

ntfHndtdx

n

ntf

n

nt

xn

nt

txnWH

RN

k

k

k

b

aN

Now combining the estimates of 6

R and ,7

R we get .)[0,

1

1)(

4

L

kfHnR

By using (11) and [3], we get

dtdxxn

ntxftxnW

iR

ik

ib

a

k

i

k

22

)(

0

2

2

12

0=

2)(2

5)(),,(

!

1

)2

(1

)(

12

0=

1)(

IL

i

k

i

kfHn

.)

2(

1

1)(2

)2

(1

1)(

IL

k

IL

kffHn

From above estimates of 4

R and ,5

R we get

.,1,)

2(

1

1)(2

)[0,1

1)(

)2

(1

,,

IL

k

L

k

IL

nffHnx

n

ntx

n

ntFS

Hence, we obtain

.)

2(

1

1)(2

)[0,1

1)(

)2

(1

3

IL

k

L

k

ILffHnR

Finally combining the estimates of ,,,321

RRR we obtain the required theorem.

Theorem 3. Let 1,),[0, pLfp

then for n sufficiently large

,),,,(,.),()[0,

1)(

1

1/2

22)

2(

,,

p

L

k

kI

pL

nfnIpnfuHfkfS

where the constant H is independent of n and .f

Proof: Let )(2,2

tfk

be the Steklov mean of th

k 2)(2 order corresponding to )( tf where

0> is sufficiently small and )( tf is defined as zero outside ),[0, then we have

)2

(2,2,,

)2

(,,

,.),(,.),(I

pL

knI

pL

nkffSfkfS

)

2(

2,2)

2(

2,22,2,,,.),(

Ip

Lk

Ip

Lkkn

fffkffS

.=:321



To estimate ,1

let )( t be the characteristic function of ,3

I then

.=:,)(=,)(542,2,,2,2,,

x

n

ntff

n

ntSx

n

ntffS

knkn

Next is true for 1,=p and 1>p according to Holder’s inequality

.)(),,(2,2

3

3

2

2

4

2

2

dxdtn

ntfftxnWdt

p

k

b

a

b

a

pb

a

Applying Fubini’s theorem, we get

.)(),,(2,2

3

3

2

2

4

2

2

dxdtn

ntfftxnWdt

p

k

b

a

b

a

pb

a

.)

3(

2,2

p

Ip

Lk

ff

Hence

.)

3(

2,2)2

(4

p

Ip

Lk

p

Ip

Lff

Applying Holder’inequality, (11) and Fubini’s theorem , for 1p we get the results.

.)[0,

2,2

1)(

)2

(5

pL

k

k

Ip

LffHn

By using Jenson’s inequality and Fubini’s theorem, we obtain

.)[0,)[0,

2,2

p

Lp

Lk

fHf

Hence

.)[0,

1)(

)[0,5

pL

k

pL

fHn

Now using rd

3 property of Steklov mean, we get

.),,,()[0,

1)(

1221

p

L

k

kfnIpnfuH

We know that.

.=)

3(

1

1)(2

2,2)

3(

1)(2

2,2IL

k

kIBV

k

kff

According to Theorem 1, Theorem 2 and Lemma 3, we have

)[0,2,2

)3

(

2)(2

2,2

1)(

2

pL

kI

pL

k

k

kffHn

,),,.()[0,

1)(

122

2)(2

pL

k

k

kfnIpnfuH

To estimate ,3

we use the rd

3 property of Steklov mean, and obtain that s

).,,,(1223

IpnfHuk

which is the required result and completes the proof of above theorem.



4. CONCLUSION

The modification of operators plays an important role in approximation theory to obtain better

approximation. In this paper, we present a direct theorem for the linear combination of stancu

type operators, we use the technique of linear approximation method.

5. ACKNOWLEDGEMENTS

The authors are thankful to the reviewer s for valuable suggestions leading to the overall

improvements in the paper.

REFERENCES

[1] Bykyazici I., Atakut C., On Stancu type generalization of q -Baskakov Operators, Math.

Comput. Modelling 52 (5-6), (2010), 752-759.

[2] Gupta V., A note on modified Baskakov type operators, Approx. Theory and its Appl, 10

(3), (1994), 74-78.

[3] Goldberg S., Meir A., Minimum moduli of ordinary differential operators, Proc. London

Math. Soc. 3, 23(1971), 1-15.

[4] Gupta V., and Srivastava G. S., Simultaneous approximation by Baskakov Szasz type

operators, Bull Math. Soc. Sci. Math. R. S. Roumanie 37 (85), (1993), no. 3-4.

[5] Hewitt E., Stromberg K., Real and abstract analysis. A modern treatment of the theory of

function of a real variable, Springer-Verlag, New York, (1965).

[6] May C. P., Saturation and inverse theorem for the combinations of class exponential type

operators, Canad. J. Math. 28 (1976), 1224-1250.

[7] Maheshwari P., Some direct estimates for Szasz- Beta operators in Lp-norm, Pure and Appl.

Mathematics Sciences 3, no. 1(2006), 61-72.

[8] Maheshwari P., Sharma D., Approximation by q -Baskakov - Beta stancu operators, Rend.

circ. Mat. Palermo 61 (2012), 297-305.

[9] Stancu D. D., Approximation of function by mean of a new generalized Bernstein operators,

Calcolo 20 (1983), 211-229.

[10] Timan A. F., Theory of approximation of functions of a real variable, International Series

of Monographs in Pure and Applied Mathematics, 34. A Pergamon Press book The

Macmillan, New York, (1963).

[11] Wood B., pL

- Approximation of linear combination of integral Bernstein type operators,

Anal. Number. Theor. Approx. 13 (1), (1984), 65-72.

[12] Zygmund A., Trignonetrical series, Dover publ. Inc. N. Y. (1995).

AUTHORS’ BIBLIOGRAPHY

Rupa Sharma

Research Scholar, Mewar University

Chittorgarh (Rajasthan), India

E-mail: [email protected]

Prerna M. Sharma

Department of Mathematics

SRM University, NCR Campus,

Modinagar (U.P), India

E-mail: [email protected]

mailto:[email protected]

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Approximation by Some Linear Positive Operators in Lp Spaces · 2015. 1. 25. · -analogue of classical Baskakov operators. In the recent years for similar type of operators some