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Generalized matrix summability of a conjugate derived Fourier series
- M Mursaleen^{1, 2}Email author and
- Abdullah Alotaibi^{2}
https://doi.org/10.1186/s13660-017-1542-5
© The Author(s) 2017
- Received: 10 August 2017
- Accepted: 12 October 2017
- Published: 2 November 2017
Abstract
The study of infinite matrices is important in the theory of summability and in approximation. In particular, Toeplitz matrices or regular matrices and almost regular matrices have been very useful in this context. In this paper, we propose to use a more general matrix method to obtain necessary and sufficient conditions to sum the conjugate derived Fourier series.
Keywords
- Fourier series
- conjugate Fourier series
- conjugate derived Fourier series
- \((E,q)\)-summability
- \((E,q)(\bar{N},p_{n})\)-summability
- \(F_{\mathfrak{B}}\)-convergence
MSC
- 42A24
- 40G05
1 Introduction
The idea of \(\mathfrak{B}\)-summability (or \(F_{\mathfrak{B}}\)-convergence) was introduced by Bell [1] and Steiglitz [2]. It generalizes the notions of A-summability and almost convergence.
Let \(\mathfrak{B}=(B_{i})_{i=1}^{\infty}\) be a sequence of infinite matrices with \(B_{i}=(b_{nk}(i))_{n,k=1}^{\infty}\). Then a bounded sequence \(x=(x_{k})_{i=1}^{\infty}\) is said to be \(\mathfrak{B}\)-summable (or \(F_{\mathfrak{B}}\)-convergent) to the value L if \(\lim_{n}(B_{i}x)_{n}=\lim_{n}\sum_{k}b_{nk}(i)x_{k}=L\), uniformly in \(i\geq 0\). In this case, L is denoted \(\mathfrak{B}\)-limx.
In this paper, we use such type of matrices, which have many applications in various fields, to study the summability problem of the conjugate derived Fourier series.
In this paper, we apply the notion of \(F_{\mathfrak{B}}\)-convergence to study the summability problems of the conjugate derived Fourier series (1.4).
2 Preliminaries
- (i)
\(\Vert A \Vert =\sup_{n}\sum_{k}\mid a_{nk}\mid<\infty\),
- (ii)
\(\lim_{n}a_{nk}=0\) for each k,
- (iii)
\(\lim_{n}\sum_{k}b_{nk}=1\).
Theorem A
- (i)
\(\Vert\mathfrak{B}\Vert<\infty\),
- (ii)
\(\lim_{n}b_{nk}(i)=0\) for all \(k\geq1\) uniformly in i, and
- (iii)
\(\lim_{n}\sum_{k}b_{nk}(i)=1\) uniformly in i,
We also need the following result on the weak convergence of sequences in the Banach space of all continuous functions defined on a finite closed interval (see [7, 8]).
Lemma B
\(\lim_{n\rightarrow\infty}\int_{0}^{\pi}g_{n}\,dh_{x}=0\) for all \(h_{x}\in BV[0,\pi]\) if and only if \(\Vert g_{n} \Vert <\infty\) for all n and \(\lim_{n\rightarrow\infty}g_{n}=0\).
We need the following well-known Dirichlet-Jordan criterion for Fourier series (see [8, 9]).
Lemma C
The trigonometric Fourier series of a 2π-periodic function f of bounded variation converges to \([f(x+0)-f(x-0)]/2\) for every x, and this convergence is uniform on every closed interval on which f is continuous.
3 Main results
Theorem 3.1
Proof
Since condition (3.9) is satisfied by condition (i) of Theorem A, it follows that (3.8) holds if and only if (3.2) holds.
This completes the proof of the theorem. □
Theorem 3.2
Let \(f(x)\) be a periodic function with period 2π and Lebesgue-integrable over \([-\pi,\pi]\). Then, for every \(x\in{}[-\pi,\pi]\) for which \(h_{x}(t)\) is continuous and of bounded variation on \([0,\pi]\), \((\tilde{S}_{k}^{\prime}(x))\) is \((E,q)\)-summable to \(\frac{1}{4\pi}\int_{0}^{\pi}\) cosec \(^{2}\frac{t}{2}\phi _{x}(t)\,dt-h_{x}(0+)\) if and only if \((\cos(k+\frac{1}{2})t)_{k}\) is \((E,q)\)-summable to 0 for every \(t\in{}[0,\pi]\).
Proof
Theorem 3.3
Let \(f(x)\) be a periodic function with period 2π and Lebesgue-integrable over \([-\pi,\pi]\). Then, for every \(x\in{}[-\pi,\pi]\) for which \(h_{x}(t)\) is continuous and of bounded variation on \([0,\pi]\), \((\tilde{S}_{k}^{\prime}(x))\) is \((E,q)(\bar{N},p_{n})\)-summable to \(\frac{1}{4\pi}\int_{0}^{\pi }\operatorname{cosec}^{2}\frac{t}{2}\phi_{x}(t)\,dt-h_{x}(0+)\) if and only if \((\cos(k+\frac{1}{2})t)_{k}\) is \((E,q)(\bar{N},p_{n})\)-summable to 0 for every \(t\in{}[0,\pi]\).
Proof
4 Conclusion
In this paper, we have characterized the \(F_{\mathfrak{B}}\)-convergence of conjugate derived Fourier series. We deduced special cases to obtain necessary and sufficient conditions for \((E,q)\)-summability and \((E,q)(\bar{N},p_{n})\)-summability of the conjugate derived Fourier series.
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. G-307-130-38. Therefore the authors acknowledge with thanks DSR for technical and financial support.
Authors’ contributions
Both authors have read the manuscript and agreed to its content and are responsible for all aspects of the accuracy and integrity of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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