#### Abstract

Recently, Alghamdi and Mursaleen (2013) used the Hankel matrix to determine the necessary and suffcient condition to find the sum of the Walsh-Fourier series. In this paper, we propose to use the Hankel matrix as well as any general nonnegative regular matrix to obtain the necessary and sufficient conditions to sum the derived Fourier series and conjugate Fourier series.

#### 1. Introduction and Preliminaries

Let and be two sequence spaces and let be an infinite matrix of real or complex numbers. We write provided that converges for each . A sequence is said to be -summable to if . If implies that , then we say that defines a matrix transformation from into and by we denote the class of such matrices. If and are equipped with the limits - and -, respectively, and -- for all , then we say that is a regular map from into and in this case we write . The matrices are called *regular, *where denotes the space of all convergent sequences.

The following are well-known Silverman-Töeplitz conditions for regularity of .

Lemma 1. * is regular if and only if*(i)*,
*(ii)* for each ,*(iii)*. *

A *Hankel matrix* is a special case of the regular matrix; that is, if then the matrix is known as the Hankel matrix. That is, a Hankel matrix is a square matrix (finite or infinite), constant on each diagonal orthogonal to the main diagonal. Its entry is a function of . The Hankel transform of the sequence is defined as the sequence , where provided that the series converges for each . An operator which transforms into as described is called the operator induced by the Hankel matrix . In [1] we can find the applications of Hankel operators to approximation theory, prediction theory, and linear system theory. Hankel matrices have a number of applications in various fields.

Recently, Al-Homidan [2] proved that Hankel matrices are regular and obtained the sum of the conjugate Fourier series under certain conditions on the entries of Hankel matrix. Most recently, Alghamdi and Mursaleen [3] proved that Hankel matrices are strongly regular. Strongly regular matrices are those matrices which transform almost convergent sequences into convergent sequences leaving the limit invariant [4].

Our aim here is to find necessary and sufficient conditions for Hankel matrix as well as any arbitrary nonnegative regular matrix to sum the derived Fourier series and conjugate Fourier series.

#### 2. Main Results

Let be -integrable and periodic with period , and let the Fourier series of be Then the series conjugated to it is and the derived series is

Let , and denote the partial sums of series (1), (2), and (3) respectively. We write where .

We propose to prove the following results.

Theorem 2. *Let be a function integrable in the sense of Lebesgue in and periodic with period . Let be a Hankel matrix. Then for each , the Hankel matrix transform of the sequence is ; that is,
**
if and only if
**
for every , where denotes the set of all functions of bounded variations on .*

In the next result, we replace the Hankel matrix by an arbitrary nonnegative regular matrix in the result of Al-Homidan [2].

Theorem 3. *Let be a function integrable in the sense of Lebesgue in and periodic with period . Let be a nonnegative regular matrix. Then -transform of the sequence converges to ; that is
**
if and only if
**
for every , where each .*

#### 3. Proofs

We will need the following lemma which is known as the Banach Weak Convergence Theorem [5].

Lemma 4. * for all if and only if for all and .*

*Proof of Theorem 2. *We have
where
Then
where
Since is of bounded variation on and as has also the same property. Hence by Jordan’s convergence criterion for Fourier series as .

Since the Hankel matrix is regular, we have
Now, it is enough to show that (6) holds if and only if
Hence, by Lemma 4, it follows that (14) holds if and only if
and (6) holds. Since (15) is satisfied by Lemma 1(i), it follows that (14) holds if and only if (6) holds. Hence the result follows immediately.

*Proof of Theorem 3. *We have
Therefore
where
Now, taking limit as on both sides of (17) and using Lemmas 1 and 4 as in the proof of Theorem 2, we get the required result.

*Remark 5. *If we take , then Theorem 2 is reduced to Theorem 4.1 of [2].

*Remark 6. *If we replace the matrix by an arbitrary nonnegative regular matrix in Theorem 2, we get Theorem 1 of Rao [6].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 130-072-D1434. The authors, therefore, acknowledge with thanks DSR technical and financial support.