On Zygmund's type strong summation theorem (Q2276711)

The main result proved is the following: Let \(f\in L\) be \(2\pi\)-periodic and let \(s_ m(f;x)\) denote the partial sum of the Fourier series of f at a point x. If for a certain x, \[ \lim_{\lambda \to 0+}\frac{1}{\lambda \eta^{p-1}}\int^{\lambda}_{-\lambda}| f(x+u)-f(x)| (\int^{u+\eta}_{u-\eta}| f(x+v)-f(x)| dv)^{p-1}du=0, \] uniformly in \(\eta >0\), with some p in \(1<p\leq 2\), then \[ \lim_{n\to \infty}(\log \frac{2n}{\nu})^{- 1}\{\frac{1}{\nu}\sum^{\nu}_{i=1}| s_{k_ i}(f;x)-f(x)|^ q\}^{1/q}=0 \] for \(q=p(p-1)^{-1}\), where \(n\leq k_ 1<k_ 2<...k_{\nu}\leq 2n\) \((n=3,4,...)\) are arbitrary indices.