The absolute generalized harmonic-Cesàro summability of a Fourier series (Q1343571)

Let \(f(t)\) be a \(2\pi\) periodic function and let \(f\in L(-\pi, \pi)\). Let \(f(x)\sim \sum^ \infty_ 0 A_ n(x)\). For fixed \(x\) and \(s\) we write \(\varphi(t)= {1\over 2}\{f(x+ t)+ f(x- t)- 2s\}\) and \[ \varphi_ \alpha(t)= {\alpha\over t^ \alpha} \int^ t_ 0 (t- u)^{\alpha- 1} \varphi(u)du,\quad \alpha> 0, \] with \(\varphi_ 0(t)= \varphi(t)\). Let \(| z,\alpha,\beta|\) denote \(| N, p_ n|\), where \(p_ n= A^{\alpha-1,\beta}_ n\) is determined by the identity: \[ (1- z)^{- \alpha- 1} (\log a/(1-z))^ \beta= \sum^ \infty_ 0 A^{\alpha,\beta}_ n z^ n,\quad | z|< 1, \] \(a> 2\). The authors prove the following theorem. Theorem: Let \(\alpha\geq 0\) and \(\beta> 1\). If \(\varphi_ \alpha(t)\log{k\over t}\in \text{BV}\), then \(\sum^ \infty_ 0 A_ n(x)\in | z,\alpha,\beta|\). The case \(\alpha= 0\) is due to \textit{G. Das} and \textit{P. C. Mohapatra} [Proc. Lond. Math. Soc., III. Ser. 41, 217-253 (1980; Zbl 0446.42004)].