Extensions of Móricz classes and convergence of trigonometric sine series in \(L^1\)-norm (Q2337253)

Summary: In this paper, the extensions of classes \(\widetilde{S}, \widetilde{C}\) and \(\widetilde{B} V\) are made by defining the classes \(\widetilde{S}_r, \widetilde{C}_r\) and \(\widetilde{B} V_r\), \(r = 0, 1, 2, \dots\) It is also shown that class \(\widetilde{S}_r\) is a subclass of \(\widetilde{C}_r \cap \widetilde{B} V_r\). Moreover, the results on \(L^1\)-convergence of \(r\) times differentiated trigonometric sine series have been obtained by considering the \(r^{t h}(r = 0,1,2,\dots)\) derivative of modified sine sum under the new extended class \(\widetilde{C}_r \cap \widetilde{B} V_r\).