On the absolute Cesàro summability of a series related to Walsh-Fourier series (Q1574272)

Let \(f(x)\) be a periodic function with period 1 and Lebesgue integrable over \((0,1)\). Then the Walsh-Fourier series of \(f\) is \[ f(t) \sim\sum^\infty_{k=0} c_kw_k(t) =\sum^\infty_{k=0} A_k(t). \] The author proves the following theorem: Let \(\phi(t) =f(x+t)-S\), where \(S\) is a function of \(x\). If \(\phi(t)\) is strongly differentiable in \(x\) and \(\int^1_0 |\phi^{(1)} (t)|/ t^\alpha dt< \infty\) for \(0<\alpha<1\), then the series \(\sum^\infty_{ n=1} n^\alpha A_n(x)\) is summable \(|c,\beta|\) for \(\beta> \alpha\).