Let \(1 0\) there exists \(\delta> 0\) such that \[ \int^\delta_0 \Biggl|\sum^\infty_{k= n} D^{(r)}_k(x)\Delta a_k\Biggr|dx< \varepsilon\quad\text{for all }n, \] where \(D^{(r)}_k(x)\) is the \(r\)th derivative of the Dirichlet kernel. This \(C_r\) is an extension of the class indicated in the title. The author proves that \(S_{pr}\) is a subclass of \(\text{BV}\cap C_r\), where BV is the class of null sequences of bounded variation. These classes play an important role in the study of the problem of \(L^1\)-convergence of the cosine series \(a_0/2+ \sum^\infty_{k=1} a_k\cos kx\).