International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 989865 | 8 pages |

On ( 𝑁 , 𝑝 , π‘ž ) ( 𝐸 , 1 ) Summability of Fourier Series

Academic Editor: HΓΌseyin Bor
Received07 Nov 2008
Revised17 Mar 2009
Accepted30 Mar 2009
Published08 Jun 2009


A new theorem on summability of Fourier series has been established.

1. Introduction

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

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

We write

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

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

The necessary and sufficient conditions for method to be regular are

and for every fixed , for which .


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

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
We use the following notations:

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 Let be a positive, nondecreasing function of . If then a sufficient condition that the Fourier Series (1.8) be summable to at the point is

3. Lemmas

Proof of the theorem needs some lemmas.

Lemma 3.1. For


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

Proof. Now considering first term of (3.4), we have Now considering second term of (3.4) and using Abel’s lemma, we have 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

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

Therefore transform of is given by We have

Now we consider

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

This completes the proof of the theorem.

5. Corollaries

Following corollaries can be derived from our main theorem.

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

Corollary 5.2. If 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


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.


  1. D. Borwein, β€œOn product of sequences,” Journal of the London Mathematical Society, vol. 33, pp. 352–357, 1958. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  2. G. H. Hardy, Divergent Series, Oxford University Press, Oxford, UK, 1st edition, 1949. View at: Zentralblatt MATH | MathSciNet
  3. A. Zygmund, Trigonometric Series. Vol. I, Cambridge University Press, Cambridge, UK, 2nd edition, 1959. View at: Zentralblatt MATH | MathSciNet

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.

More related articles

676Β Views | 437Β Downloads | 0Β Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.