• No results found

Approximation of Conjugate Function Belonging to Lip(ξ(t),r) Class by (C,1)(E,q) Means

N/A
N/A
Protected

Academic year: 2020

Share "Approximation of Conjugate Function Belonging to Lip(ξ(t),r) Class by (C,1)(E,q) Means"

Copied!
10
0
0

Loading.... (view fulltext now)

Full text

(1)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

162

Approximation of Conjugate Function Belonging to Lip(ξ(t),r)

Class by (C,1)(E,q) Means

Kalpana saxena

1

, Sheela Verma

2

1,2Department of Mathematics, Govt. M.V.M. Barkatullah University, Bhopal India

Abstract:- In this paper, introduce the concept of (C,1)(E,q) product operators and establishes a new theorems on degree of approximation of a function ̅, conjugate to a 2π periodic function f, belonging to Lip(ξ(t),r) Class by (C,1)(E,q) product means conjugate fourier series have been established.

Keywords And Phrases:- Degree of approximation, Lip(ξ(t),r) class of function, (C,1) Summability, (E,q) Summability, (C,1)(E,q) product Summability, fourier series, conjugate fourier series, Lebesgue integral.

I. INTRODUCTION

The degree of approximation of the function belonging to Lipα by Cesaro means and norlund means has been discussed by a number of researches like Alexits[6], Sahney and Goel[9], Chandra[1],\Qureshi[8], Rhoades[4], Hare Krishna Nigam[14] obtained the degree of approximation of the function belonging to Lipα by (C,1)(E,1) product means of its fourier series.

In the present we have extended this result by obtaining the degree of approximation of function f belonging to a generalized class Lipα.. Mittal and singh [7], Rhoades etal. [8], mishra etal. [9,10] and mishra and mishra [11] for function of various classes Lipα, Lip( ,r), Lip(ξ(t),r) and W(Lr,ξ(t)),(r 1) by using various Summability methods. But till now, nothing seems to have been done as far to obtain the degree of approximation of conjugate of a function (C,1)(E,q) product Summability method of its conjugate series of fourier series. In this paper, we obtain a new theorem on the degree of approximation of a function

̃, conjugate to a periodic function f Lip(ξ(t),r)- Class , by (C,1)(E,q) product Summability means.

Definition and notation:- Let f(x) be periodic with period 2π and integrable in the sense of Lebesgue. The fourier series of f(x) is given by

( ) ∑ ( ) (1.1)

With nth partial sums ( )

The conjugate series of the fourier series (1.1) is given by

( ) (1.2)

With nth partial sums ̅̅̅ (f;x) and we shall call it as conjugate fourier series through out of paper

- norm of a function f:R→R is defined by ‖ ‖ *| ( )| +

- Norm is defined by ‖ ‖ (∫ | ( )| ) , (1.3)

The degree of approximation of a function f:R→R by a trigonometric polynomial tn of order n under Sup norm ‖ ‖ is

defined by Zygmund [13] and is given as

‖ ‖ *| ( ) ( )| + (1.4)

And ( ) is given by

( ) ‖ ‖ (1.5)

(2)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

163

f(x+t) ― f(x) = O(| | ) for 0 1 (1.6)

f(x) ( ) if

Let ∑ be a given infinite series with the sequence of its nth partial sums * +

The (C,1) transform is defined as the nth partial sum of (C,1) Summability

∑ n (1.10)

Then the series ∑ is summable to S by (C, 1) method

( ) ( ) {∑ ( ) } n (1.11)

Then the infinite series ∑ is said to be summable (E,q) to the definite number S (Hardy[3]),

The (C,1) transform of the (E,q) transform defines (C,1)(E,q) transform of the partial sum of series ∑ and we denote

it by ( ) .

Thus if ( )

=

∑ *( ) ∑ ( ) +

Where denotes the (E,q) of and denotes (C,1) transform of , then the series ∑ is said to be Summable

to(C,1)(E,q) method or summable (C,1)(E,q) to a definite number S.

We use the following notations:

( ) ( ) ( )

= Integral part of * +

̅ ( )

( )∑ [( ) ∑ ( )

( )

]

II. MAIN THEOREMS

A good amount of work has been done on degree of approximation of functions belonging to Lip ( ) (ξ( ) ) classes using Cesaro, N ̈rlund and generalized N ̈rlund single Summability methods by a number of researchers like Alexits [6], Sahney and Goel DS [9], Qureshi [8], Chandra [1] and Rhoades[4].

But till now nothing seems to have been done so far in the direction of present work. Therefore ,in present paper, a theorem on degree of approximation of a function ̅, conjugate to periodic function f, belonging to

(3)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

162

2.1 Theorem:-

If f is a -periodic function, Lebesgue integrable on [0,2π], belonging to the (ξ( ) ) class then its degree of approximation by ( )( )̅̅̅̅̅̅̅̅̅̅̅̅̅̅ summability means of its conjugate fourier series is given by

‖( ̅̅̅̅) ‖ *( ) ( )+ (2.1)

Provided ξ(t) satisfies the following condition:

{∫ ( | ( )| ( ))

} (

) (2.2)

And

{∫ ( | ( )| ( ) )

} {( ) }

(2.3)

Where is an arbitrary number such that

Condition (2.2) and (2.3) hold uniformly in and ̅̅̅̅̅̅ ( )( )̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ means of the series (1.2) provided

( ) ( ( ) ) (2.4)

And ̅( )

∫ ( )

III. LEMMAS

For the proof of our theorem following lemmas are required:

Lemma 1:̅̅̅( ) ( )

Proof: For

( ) ( ) | |

| ̅̅̅( )|

( )|∑ [( ) ∑ ( )

( )

( )

]

|

( )∑ [( ) ∑ ( )

]

( )∑ [( ) ∑ ( )

]

= ( )

(4)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

163

Lemma-2For and any n, we have

̅̅̅( ) .

( )/ (( )( ) ∑( )

)

Proof: For ( ⁄ ) ( ⁄ )

| ̅̅̅( )|

( )|∑ [( ) ∑ ( )

( )

( )

]

|

( )|∑ [( ) {∑ ( )

( )

}]

|

( )|∑ [( ) {∑ ( )

}]

| | |

( )|∑ [( ) {∑ ( )

}]

|

( )|∑ [( ) {∑ ( )

}]

|

( )|∑ [( ) {∑ ( )

}]

|

Now,

| |

( )|∑ [( ) {∑ ( )

}]

|

( )|∑ [( ) {∑ ( )

}]

| | |

( )∑ [( ) {∑ ( )

}]

( )∑

(5)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

164

( )

(

( )) (3.2)

Now

Considering second form of (3.1) and using Abel’s Lemma

| |

( )|∑ [( ) {∑ ( )

}]

|

( )∑ ( ) |∑ ( )

|

( )( ) ∑( )

( )( ) ∑ ( ) ⌉ (3.3)

Combining (3.1),(3.2) and (3.3), we get

̅̅̅( ) (

( )) (( )( ) ∑ ( )

) (3.4)

IV. PROOF OF THE THEOREM:-

Let ̅ ( ) denote S, the nth partial sum of the series (1.2). Then following Lal[5], we have,

̅ ( ) ̅( )

∫ ( )

( )

Therefore using (1.2). the (E,q) transform of ̅ ( ) is given by

̅̅̅̅ ̅( )

( ) ∫ ( ) {∑ ( )

( )

( )

}

Now denoting ( )( )̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ transform of ̅ by ( ) ,

We write

( )

̅̅̅̅̅̅̅̅ ̅( )

( )∑( )

∫ ( )

[∑ ( )

( )

(6)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

165

∫ ( )̅̅̅( )

Where

̅̅̅( )

( )∑ [( ) ∑ ( )

( )

( ) ]

[∫ ∫

] ( ) ̅̅̅( )

(say) (4.1)

We consider,

| | ∫ | ( )| ̅̅̅( )

Using H ̈lders’s inequality and the fact that

( ) ( ( ) )

Approximation of conjugate function belonging to ( ( ) ) class

[∫ 2 | ( )|

( ) 3

] [∫ 2 ( )| ̅̅̅( )| 3 ]

(

) [∫ ,

( )| ̅̅̅̅( )|

-

] by (2.2)

( ) [∫ , ( )- ] by lemma-1

Since ( )is a positive increasing function and using second mean value theorem for integrals,

,( ) ( )- [∫ ] for some

= ,(

) ( )- 0,

1

= O,(

(7)

-International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

166

= O,( ) (

)- (4.2)

Using H ̈lder’s inequality,

| | ∫ | ( )|| ̅ ( )|

dt

| | [∫ | ( )| ( )

]

+O[∫ | ( )|

( )( )

∑ ( )

] (by lemma 2) (4.3)

=O( ) ( ) say

Using H ̈lder’s inequality and lemma 2

| | ( ) [∫ 2

( )

( ) 3

] [∫ 2 ( ) 3

]

= O,( ) - [∫ , ( ) -

] by (2.3)

Now putting t=

2

( )

3 [∫ {

( ) ( ) }

]

Since ( )is a positive increasing function and using second mean value theorem

For integrals,

,( ) ( )- *∫ ( )

+ for some

= O,( )

( )- *∫

( )

+ for some

= O,( )

( )- 0{

( )

( ) }

1

= O,( )

( )- *( ) ( ) +

{ (

)} {( )

}

,( ) (

(8)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

167

0

( ) 1 (4.4)

Similarly

| | ( ) [∫ 2

( )

( ) 3

] [∫ 2 ( )( ) ∑ ( )

3

]

= O( ) [∫ , ( ) ( )

] [∫ ,

( )( )

-

] by 2.4

= O[∫ , ( ) ( )

] [∫ ,

( )

-

]

= O{( ) } [∫ , ( ) -

]

by (2.3)

Now putting t=

{( ) } [∫ {

( ) ( ) }

]

Since ( )is a positive increasing function and using second mean value theorem

For integrals,

,( ) ( )- *∫ ( )

+ for some

= ,( ) ( )- *∫

( )

+ for some

= ,( ) ( )- 0{

( )

( ) }

1

{( ) (

)} [( )( ) ]

{ (

)} [( )

]

(9)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

168

Combining (4.1) (4.2) (4.3) (4.4) & (4.5)

|( )̅̅̅̅̅̅̅̅ ̅( )| {( ) ( )}

Now Using - norm we get,

‖( ̅̅̅̅) ( )‖ 2∫ ‖( ̅̅̅̅) ( )‖

3

= [{∫ ,( ) (

)-

} ]

,( ) ( )- 0,∫ - 1

= ,( ) (

)-This completes the proof of theorem.

Applications:- The following corollaries can be derived from our main theorem:

Corollary 1: If ( ) = , 0 1 , then the class Lip( ( ) ), r 1,

Reduces to the class Lip( ) and the degree of approximation of the function ̅, conjugate to a periodic function f

Lip( ), 1 , is given by|( )̅̅̅̅̅̅̅̅ ̅| .

( ) /

Proof:- ‖( ̅̅̅̅) ̅‖ ,∫ ‖( ̅̅̅̅) ̅‖

-Or

,( ) ( )- = ,∫ ‖( ̅̅̅̅) ̅‖

-Or

O(1) = ,∫ ‖( ̅̅̅̅) ̅‖ - .O{

( ) ( )}

Hence

(10)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 1, January 2017)

169

For if not the right- hand side will be O(1) , therefore

|( )̅̅̅̅̅̅̅̅ ̅| {(

) ( ) }

= O.

( ) /

Corollary 2:- If r in Corollary 1, then the class Lip( ) reduces to the Lip class and the degree of approximation of a

function ̅, Conjugate to a periodic function f Lip , 0 1 is given by

‖( ̅̅̅̅) ̅‖ O( ( ) )

Remark:- An independent proof of above corollaries 1 can be obtained along the same lines of our theorem.

Acknowledgement

Author is thankful to her family their encouragement and support to this work.

REFERENCES

[1] Chandra, Holder degree of Approximation of function in the metric, Journal of the Indian Maths Soc. Vol. 53(1988) . 99-114

[2] Lal S. and kushwaha J. Degree of approximation of Lipschitz function by product Summability method, Int. Math. Forum, 4, 2009, No. 43, 2101-2107.

[3] Rhoades B.E, On degree of approximation of a function belonging to Lipschitz class, (2003).

[4] Zygmund A. Trignometrical series, Cambridge University Press(1960)

[5] Alexits G., Uber die annaherung einer metigen function durch die Cesaro Schen mittle ihrer Fourier-reihe, Math. Ann., 100(1928), 264-277.

[6] Qureshi K., On degree of approximation of a function belonging to the Lip class, Indian J. of Pure & Applied maths, 13(1982), 898. [7] Sahney B.N. and God D.S. On degree of approximation of a

function, Ranchi Univ., Math J. 4(1973).

[8] Approximation of Conjugate Function belonging to Lip( ( ) ) Class by (C,1)(E,1) Means.

BIOGRAPHY

Dr. Kalpana Saxena received Ph.D. in Mathematics from University of APS Rewa 1999. Dr. Kalpana Saxena is presently posted as a professor at GOVT. M.V.M. College Bhopal. Her main interests are Product Summability of Fourier series and sequence.

Sheela Verma perusing Ph.D. in

References

Related documents

The final reading approval of the thesis was granted by Martin Corless-Smith, Ph.D., Chair of the Supervisory Committee.. The thesis was approved for the Graduate College

Jefimenko derived the equations of gravitodynamics in a different way, starting from make an analogy of his retarded solutions of the electromagnetic field to the

David Brown writes that “not only had there never before been such classes [music theory] in Russia; there was not even a textbook on harmony in Russian (it would be

Many, if not most, of the people to be displaced by the development project were tribal "encroachers" on state land because they held no legal title to that

Recently its fishery exports have been rejected by many countries because of the violation of human rights in the country especially of migrant workers who work in fishing industry..

Bergounioux, Optimal control of abstract elliptic variational inequalities with state constraints, to appear, SIAM Journal on Control and Optimiza- tion , Vol. Tiba, General

Human resources are strategic partners to ensure and create organizational diagnoses by implementing steps in the organization, namely: establishing a system that underlies

Diff erent strains of rats are often used to elucidate the pathogenesis of neuropathic pain, and due to the inad- equate test procedures researchers often reported strain-