Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 42, 2081 - 2087

Approximation of Conjugate

Functions by their Fourier Series

Ratna Singh

Department of Applied Mathematics

Gyan Ganga College of Technology, Jabalpur-482003, India

ratnaverma8@gmail.com

S. S. Thakur

Department of Applied Mathematics, Jabalpur Engineering College

Jabalpur-482011, India

samajh_singh@rediffmail.com

Abstract

In this paper, we take up generalized Nörlund mean

to study the

degree of approximation of ̃ , the conjugate function of , ,

under the sup – norm. The particular cases of our theorems provide

the degree of approximation of ̃ by ( )-mean and ( )–mean of Fourier

series of ̃ which yield the Jackson order for .

Keywords: Rate of convergence, degree of approximation, Trigonometric

approximation

1. DEFINITION AND NOTATIONS

Let [ ]be 2π - periodic and let theFourier series of at be given

by

( )

∑( )

Then its conjugate series at is given by

( ) ∑( )

We write

2082 R. Singh and S. S. Thakur

( ) ̃( )

∫{ ( ) ( )} ( ⁄ )

whenever it exists. The function ̃is called conjugate function of . We denote by

( )and ( ̃ )respectively the partial sums of the series (1.1) and (1.2).

Let ( ) and ( ) be non-negative such that and for all . Then the generalized Nörlund mean ( ) of

a real valued sequence ( ) is given by

( ) ( )

∑

The mean

reduces to Nörlund ( ) and Riesz ( ) means respectively on letting and ( ) in (1.4) (see

[4,5] ). In special cases ( )-mean reduces to Harmonic and Cesàro means (see Hardy [6]; p-110 and Boos [2], p-127).

Throughout ‖ ‖ will be the sup- norm with respect to x for

We also use ( or ) for non- decreasing ( or non- increasing).

We write

( ) ( ) ( ) ( ).

2. INTRODUCTION

In 1973, Sahney and Goel [8] obtained the following result using Nörlund

mean:

THEOREM A. Let , , then the degree of

approximation of by ( )-mean of its Fourier series is given by

( ) ‖ ( ) ‖ {( ) ∑

}

where ( ) are the Nörlund mean of the Fourier series of and ( ) is

positive and .

The following theorem concerning the degree of approximation of

continuous functions by the Riesz mean is due to Chandra [3]:

THEOREM B. Let , and let ( ) . Then the

degree of approximation of by ( ) -mean of its Fourier series is given by

Approximation of conjugate functions 2083

( ) ‖ ( ) ‖ { {( ⁄ ) ( ⁄ )}

{( ⁄ ) }

where ( ) are the ( ) –mean of the Fourier series of .

In the case when ( ) , the following result is due to Singh [9]:

THEOREM C. Let , and let ( ) be positive and

sequence such that

( ) ∑| | ( )

Then the degree of approximation of by ( )-means of its Fourier series is

given by

( ) ‖ ( ) ‖ { {( ⁄ ) ( ⁄ )}

{( ⁄ ) }

The object of this paper is to obtain the degree of approximation of ̃ (i.e.

the conjugate function of ) by using generalized Nörlund mean

. Precisely

we prove the following:

THEOREM 1. Let , and let ( ) and ( ) be

non-negative sequences respectively. Then the degree of approximation of ̃ by ( )- means of its Fourier series is given by

(2.5) ‖ ( ̃) ̃‖ {

{( ⁄ ) ( ⁄ )}

{( ⁄ ) }

where ( ̃ ) are generalized Nörlund means of the Fourier series of ̃ at .

THEOREM 2.Let , and let ( ) and ( ) be

non-negative sequences respectively and let ⁄ be . Then the degree of

approximation of ̃ by ( )- means of its Fourier series is given by

( ) ‖ ( ̃) ̃‖ {( )

∑

}

3. LEMMAS

We shall use the following lemmas in the proof of the theorems:

LEMMA1.[1,p-99]. If , , then ̃ (0 ).

LEMMA2.[1,p-52]. If ̃ is integrable then the series (1.2) is its Fourier series.

2084 R. Singh and S. S. Thakur

LEMMA3.[7,10] . Let ( ) and ( ) be non-negative and sequences

respectively, then , for and for any n , we have

( ) |∑ ( )

| [ ⁄ ]

for some positive constant A.

LEMMA4. Let the sequence ( ) and ( ) and for any n. Then

( ) |∑ (

)

| [ ⁄ ]

This is a easy consequence of lemma 3.

LEMMA5. Let ( ) and ( ) , then uniformly in

( ) |∑ (

)

| ( ⁄ )

This may be obtained by using Abel’s lemma.

4. PROOF OF THE THEOREMS

PROOF OF THEOREM 1. By Lemma 2, (1.2) is the Fourier series of ̃ , the

conjugate function of defined by (1.3). We assume

(4.1) ( ̃ )= 0,

Then( )- mean of ( ̃ ) is given by

( )

( ̃ ) ( ) ∑ ( ̃ )

where

( ̃ ) ̃( )

∫ ( )

( )

Hence

( ̃ ) ̃( )

∫

( )

(∑ (

)

)

(∫ ∫

)

(4.3) , say,

where

⁄ . Then , by Minkowski’s inequality

(4.4) ‖ ( ̃) ̃‖ ‖ ‖ ‖ ‖ .

By Lemma 1 , we observe that

Approximation of conjugate functions 2085

(4.5) ‖ ( )‖ ( ) ,

and since (

) ( ) , we get by (4.5)

‖ ‖ ( )

∫

(∑

)

(4.6) {( ⁄ ) } .

And by Lemma 5 and (4.5)

‖ ‖ ( )

∫

( ⁄ )

{( ⁄ ) ∫

}

(4.7) { {( ⁄ ) ( ⁄ )}

{( ⁄ ) }

Using ‖ ‖ and ‖ ‖ in (4.4), we get

‖ ( ̃) ̃‖ {

{( ⁄ ) ( ⁄ )}

{( ⁄ ) }

This completes the proof of Theorem 1.

PROOF OF THEOREM 2. Proceeding as in Theorem 1, we get

( ̃ ) ̃( )

(∫ ∫

){ ( )

(∑ (

)

) }

(4.8) , say.

Once again, by Minkowski’s inequality

‖ ( ̃) ̃‖ ‖ ‖ ‖ ‖.

Now , by (4.5)

‖ ‖ ( )

∫

(∑

)

( )

( ) {

∑

}

since

( )

∑

∑

2086 R. Singh and S. S. Thakur

by using the fact that ⁄ is .

Also, by Lemma 4 and (4.5)

‖ ‖

∫ ‖ ( )‖ |∑ (

)

|

{

∫

[ ⁄ ] }

{

∑ ∫

( ) }

( ) {

∑

}

Combining ‖ ‖ and ‖ ‖, we get the proof of the theorem.

5. APPLICATIONS

We now apply these general results to obtain some of the new results on the

degree of approximation.

Putting , for all , in Theorem1, we get

COROLLARY 1.Let , and let ( ) and ( ) be

positive and . Then the degree of approximation of ̃ by ( )- means of its

Fourier series is given by

( ) ‖

( ̃) ̃‖ { {( ⁄ ) ( ⁄ )}

{( ⁄ ) }

where ( ̃ ) are the ( )- means of the Fourier series of ̃.

The case , for all to Theorem 2 yields the following new result:

COROLLARY 2. Let , and ( ) . Then the

degree of approximation of ̃ by ( )- means of its Fourier series is given by

( ) ‖ ( ̃) ̃‖ {( )

∑

}

If we put , for all , in our Theorems 1 and 2, we may get

analogous results for ( ) – mean of the Fourier series of ̃ .

Approximation of conjugate functions 2087

References

[1] N.K.Bary, A Treatise on trigonometric series Vol. I, Pergamonpress, 1964.

[2] J. Boos, Classical and modern methods in summability, Oxford Univ.

Press, 2000.

[3] P. Chandra, On the degree of approximation of functions belonging to

Lipschitz class , Nanta Math., 8(1975), 88-91.

[4] Ming-Po Chen, On some methods of summability II, Tamkang J. Math.,

3(1972), 87- 94.

[5] D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney, Best

approximation by a saturation class of polynomial operators , Pacific J.Math.,

55(1974),149-155.

[6] G. H. Hardy, Divergent series , Oxford Univ. Press, 1949.

[7] H. H. Khan , On the degree of approximation of functions belonging to the

class ( ), Indian J. Pure Appl. Math., 5(1974), 132-136.

[8] B. N. Sahney and D. S. Goel, On the degree of approximation of

continuous functions , Ranchi Univ. Math. J., 4(1973), 50-53.

[9] T. Singh, On the degree of approximation by linear polynomial operators,

Math. Student, 47 (1979), 222-225.

[10] T.Singh and B. P. Raghuwanshi, Degree of approximation by linear

operators, Indian J. Pure Appl. Math., 21(9) (1990), 823-832.

Received: June, 2012