On L1-convergence of certain generalized modified trigonometric sums

Abstract

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