On $L^{1}$-convergence of certain generalized modified trigonometric sums

Abstract

In this paper we define new modified generalized sine sums$K_{nr}(x)=\dfrac{1}{2\sin x}\sum_{k=1}^{n}(\triangle^{r}a_{k-1}-\triangle^{r}a_{k+1})\tilde{S}_{k}^{r-1}(x)$ and study their $L^{1}$-convergence under a newly defined class $\bold{K}^{\alpha}$.Our results generalize the corresponding results of Kaur, Bhatia and Ram [6] and Kaur~[7].