Let \(\sum A_n(x)\) denote the Fourier series of \(f\in L_{2\pi}\) at a point \(x\) and let \[ 2\phi(t)= f(x+ t)+ f(x- t)- 2f(x)= 2\phi_0(t) \] and \[ \phi_\alpha(t)= \alpha t^{-\alpha} \int^t_0 (t- u)^{\alpha- 1} \phi(u)du\quad (\alpha> 0). \] Throughout, suppose \(q>0\), \(\pi\geq c>0\) and \(a\geq 0\). In this paper, the author has examined some of the cases of the following general proposition: Let, for \(\alpha\geq 0\) and \(b\leq 0\), \(a-\alpha\geq 2b\). Then \(t^{-a}\phi_\alpha(t)\in \text{BV}(0,c)\) is a sufficient condition for \(\sum A_n(x) n^b\in| E,q|\). The case \(b= 0\) and \(a=\alpha\), where \(\alpha\) is a positive integer, of the above proposition was examined for its validity by the present reviewer [Riv. Mat. Univ. Parma, IV. Ser. 3, 65-78 (1977; Zbl 0397.42004)]. In the paper under review, the author has shown that the following cases of the above proposition are valid: (1) \(\alpha>0\) and \(b=0\), (2) \(\alpha>0\) and \(b= -1\), and (3) \(0<\alpha< 2\) and \(b<0\) with \(\alpha+ b<1\). The investigations for the other cases of the above proposition are still open.