H. K. Nigam and Ajay Sharma

Copyright © 2009 H. K. Nigam and Ajay Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new theorem on (N,p,q)(E,1) summability of Fourier series has been established.

1. Introduction

Let and be the sequences of constants, real or complex, such that

(1.1) Given two sequences and convolution is defined as (1.2) Let be an infinite series with the sequence of its th partial sums .

We write (1.3)

If , for all , the generalized Nörlund transform of the sequence is the sequence .

If , then the series or sequence is summable to by generalized Nörlund method (Borwein [1]) and is denoted by

(1.4)

The necessary and sufficient conditions for method to be regular are (1.5)

and for every fixed , for which .

Now (1.6)

If then the series is said to be summable to (Hardy [2]): (1.7)

If then we say that the series or the sequence is summable to by summability method.

Particular Cases

(1) mean reduces to summability mean if . (2) mean reduces to mean if (3) method reduces to if (4) method reduces to if ,. Let be a periodic function with period and integrable in the sense of Lebesgue over the interval .

Let its Fourier series be given by

(1.8) We use the following notations: (1.9)

2. Theorem

A quite good amount of work is known for Fourier series by ordinary summability method. The purpose of this paper is to study Fourier series by summability method in the following form.

Theorem 2.1. Let be positive monotonic, nonincreasing sequences of real numbers such that (2.1) Let be a positive, nondecreasing function of . If (2.2) (2.3) then a sufficient condition that the Fourier Series (1.8) be summable to at the point is (2.4)

3. Lemmas

Proof of the theorem needs some lemmas.

Lemma 3.1. For (3.1)

Proof. (3.2)

Lemma 3.2. If and are nonnegative and nonincreasing, then for and any we have (3.3)

Proof. (3.4) Now considering first term of (3.4), we have (3.5) Now considering second term of (3.4) and using Abel’s lemma, we have (3.6) Using (3.5) and (3.6), we get the required result of Lemma 3.2.

4. Proof of Theorem

Following Zygmund [3], the th partial sum of the series (1.8) at is given by (4.1)

So the mean of the series (1.8) at is given by (4.2)

Therefore transform of is given by (4.3) We have (4.4)

Now we consider (4.5)

Now by Riemann-Lebesgue theorem and by regularity of the method of summability we have

(4.6)

This completes the proof of the theorem.

5. Corollaries

Following corollaries can be derived from our main theorem.

Corollary 5.1. If (5.1) then the Fourier series (1.8) is summable to at the point .

Corollary 5.2. If (5.2) then the Fourier series (1.8) is summable to at the point , provided that be a positive, monotonic, and nonincreasing sequence of real numbers such that (5.3)

Acknowledgments

The authors are grateful to Professor Shyam Lal, Department of Mathematics, Harcourt Butler Technological Institute, Kanpur, India, for his valuable suggestions and guidence in preparation of this paper. The authors are also thankful to Professor M. P. Jain, Vice Chancellor, Mody Institute of Technology and Science (Deemed University), Lakshmangarh, Sikar, Rajasthan, India, and to Professor S. N. Puri, Former Dean, Faculty of Engineering, Technology, Mody Institute of Technology and Science (Deemed University), Laxmangarh, Sikar, Rajasthan.

References

1. D. Borwein, “On product of sequences,” Journal of the London Mathematical Society, vol. 33, pp. 352–357, 1958.
2. G. H. Hardy, Divergent Series, Oxford University Press, Oxford, UK, 1st edition, 1949.
3. A. Zygmund, Trigonometric Series. Vol. I, Cambridge University Press, Cambridge, UK, 2nd edition, 1959.