Convergence and Summability of Fourier Sine and Cosine Series with Its Applications

Abstract

The existence of sine and cosine series as a Fourier series, their $$L^1$$ -convergence seems to be one of the prominent question in theory of convergence of trigonometric series in $$L^1$$ -metric norm. In the literature, till now most of the authors have studied the $$L^1$$ -convergence of cosine trigonometric series. However, very few of them have studied the $$L^1$$ -convergence of trigonometric sine series. In this paper, new modified cosine and sine sums of Fourier series are introduced and a criterion for the summability and $$L^1$$ -convergence of these modified sums is obtained. Also, necessary and sufficient condition for the $$L^1$$ -convergence of cosine and sine series is deduced as corollaries. Further an application is given to illustrate the main result.