1. Introduction

Let be a given infinite series with the sequence of partial sums . Let be a sequence of positive real numbers such that

(1.1)

The sequence –to-sequence transformation

(1.2)

defines the sequence of the -mean of the sequence generated by the sequence of coefficient . If

(1.3)

then the series is said to be summable to s.

The conditions for regularity of - summability are easily seen to be

(1.4)

The sequence to-sequence transformation, ^[1]

(1.5)

defines the sequence of the mean of the sequence . If

(1.6)

then the series is said to be summable to s.

Clearly method is regular. Further, the transform of the transform of is defined by

(1.7)

If

(1.8)

then is said to be -summable to s.

Let be a periodic function with period 2π and integrable in the sense of Lebesgue over (-π,π). Then the Fourier series associated with at any point x is defined by

(1.9)

and the conjugate series of the Fourier series (1.9) is

(1.10)

Let be the n-th partial sum of (1.10). The -norm of a function is defined by

(1.11)

and the -norm is defined by

(1.12)

The degree of approximation of a function by a trigonometric polynomial of degree n under norm is defined by ^[6]

(1.13)

and the degree of approximation of a function is given by

(1.14)

This method of approximation is called Trigonometric Fourier approximation.

A function if

(1.15)

and , for , if

(1.16)

For a given positive increasing function , the function , if

(1.17)

For a given positive increasing function and an integer the function , if

(1.18)

We use the following notation throughout this paper:

(1.19)

and

(1.20)

Further, the method is assumed to be regular throughout the paper.

2. Known Theorems

Dealing with the degree of approximation by the product, Misra et al ^[2] proved the following theorem using mean of conjugate series of Fourier series:

If is a periodic function of class , then degree of approximation by the product summability means of the conjugate series (1.10) of the Fourier series (1.9) is given by where is as defined in (1.7).

Recently Misra et al ^[3] established a theorem on degree of approximation by the product mean of the conjugate series of Fourier series of a function of class . They prove:

If is a periodic function of class , then degree of approximation by the product means of the conjugate series (1.10) of the Fourier series (1.9) is given by , where is as defined in (1.7).

Extending to the function of the class , very recently Misra et al ^[4] have proved a theorem on degree of approximation by the product mean of the conjugate series of the Fourier series of a function of class . They prove:

Let be a positive increasing function and a Periodic function of the class . Then degree of approximation by the product summability means on the conjugate series (1.10) of the Fourier series (1.9) is given by , where is as defined in (1.7).

Further extending to the class of functions , in the present paper, we establish the following theorem:

3. Main result

Let be a positive increasing function and a Periodic function of the class . Then degree of approximation by the product summability means on the conjugate series (1.10) of the Fourier series (1.9) is given by

(3.1.1)

provided

(3.1.2)

and

(3.1.3)

hold uniformly in with , where is an arbitrary number such that and is as defined in (1.7).

4. Required Lemmas

We require the following Lemmas to prove the theorem.

LEMMA 4.1:

Proof:

For , we have sin nt ≤ n sin t then

This proves the lemma.

LEMMA 4.2:

Proof:

For, by Jordan’s lemma, we have.

Then

This proves the lemma.

5. Proof of Main Theorem

Using Riemann–Lebesgue theorem, for the n-th partial sum of the conjugate Fourier series (1.10) ofand following Titchmarch ^[5], we have

Using (1.2), the transform of is given by

Denoting the transform of by , we have

(5.1)

Now

where , using Hölder’s inequality

(5.2)

Next

where , using Hölder’s inequality , using Lemma 4.2 and (3.1.3)

since is a positive increasing function, so is . Using second mean value theorem we get

(5.3)

Then from (5.2) and (5.3), we have

This completes the proof of the theorem.

6. Corollaries

Following corollaries can be derived from the main theorem.

Corollary 6.1: The degree of approximation of a function belonging to the class is given by

Proof: The corollary follows by putting and in the main theorem.

Corollary 6.2: The degree of approximation of a function belonging to the class is given by

Proof: The corollary follows by letting in corollary 6.1.

References

[1]	G.H. Hardy, Divergent Series (First Edition), Oxford University Press, (1970).
	In article
[2]	U.K. Misra, M. Misra, B.P. Padhy and S.K. Buxi, “On degree of approximation by product means of conjugate series of Fourier series”, International Jour. of Math. Scie. And Engg. Appls. ISSN 0973-9424, Vol 6 No. 1 (Jan. 2012), pp 363-370
	In article
[3]	Misra U.K.,Paikray, S.K., Jati, R.K, and Sahoo, N.C.: “On degree of Approximation by product means of conjugate series of Fourier series”, Bulletin of Society for Mathematical Services and Standards ISSN 2277-8020, Vol. 1 No. 4 (2012), pp 12-20.
	In article
[4]	U.K. Misra, M. Misra, B.P. Padhy and D.Bisoyi, “On Degree of Approximation of conjugate series of a Fourier series by product summability" Malaya Journal of Mathematik (ISSN: 2319-3786, Malayesia), Vol. 1 Issue 1 (2013), pp 37-42.
	In article
[5]	E.C. Titchmarch, The Theory of Functions, Oxford University Press, (1939).
	In article
[6]	A. Zygmund, Trigonometric Series (Second Edition) (Vol. I), Cambridge University Press, Cambridge, (1959).
	In article