Convergence and Summability of Fourier Sine and Cosine Series with Its Applications
AbstractThe existence of sine and cosine series as a Fourier series, their $L^1$$L1 convergence seems to be one of the prominent question in theory of convergence of trigonometric series in $L^1$$L1 metric norm. In the literature, till now most of the authors have studied the $L^1$$L1 convergence of cosine trigonometric series. However, very few of them have studied the $L^1$$L1 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$$L1 convergence of these modified sums is obtained. Also, necessary and sufficient condition for the $L^1$$L1 convergence of cosine and sine series is deduced as corollaries. Further an application is given to illustrate the main result.