Generalized matrix summability of a conjugate derived Fourier series

M Mursaleen,Abdullah Alotaibi

doi:10.1186/s13660-017-1542-5

M Mursaleen, Abdullah Alotaibi

Open Access

https://doi.org/10.1186/s13660-017-1542-5

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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.

Highlights

The idea of B-summability was introduced by Bell [ ] and Steiglitz [ ]
Let c denote the space of all convergent sequences
Lemma C The trigonometric Fourier series of a π -periodic function f of bounded variation converges to [f (x + ) – f (x – )]/ for every x, and this convergence is uniform on every closed interval on which f is continuous

Summary

Introduction

The idea of B-summability (or FB-convergence) was introduced by Bell [ ] and Steiglitz [ ]. Let B = (Bi)∞ i= be a sequence of infinite matrices with Bi = (bnk(i))∞ n,k= . A bounded sequence x = (xk)∞ i= is said to be B-summable (or FB-convergent) to the value L if limn(Bix)n = limn k bnk(i)xk = L, uniformly in i ≥ . We use such type of matrices, which have many applications in various fields, to study the summability problem of the conjugate derived Fourier series.

Results

Conclusion