On the absolute Hausdorff summability factors of the conjugate series of a Fourier series (Q1070466)

Let f(t) be a \(2\pi\)-periodic function in L(-\(\pi\),\(\pi)\), and \(\sum^{\infty}_{n=1}B_ n(\theta)\) be the series conjugate to the Fourier expansion of f. Put \(\psi (t)=\{f(\theta +t)-f(\theta - t)\}/2,\quad h(t)=\psi (t)/\log (k/t)\) \((k>\pi)\). A summation method \((H,\mu_ n)\), \(\mu_ n=\int^{1}_{0}x^ n d\chi (x)\), is said to belong to \({\mathfrak M}\) if (i) \((H,\mu_ n)\) is conservative, \((ii)\quad \int^{1}_{0}\log \log (B/(1-x))| d\chi (x)| <\infty\) \((B>e)\), \((iii)\quad \int^{1}_{0}\log (2/x)| d\chi (x)| <\infty.\) The author shows that if \(h(t)\log (k/t)\in BV(0,\pi)\), \(h(t)/t\in L(0,\pi)\), \((H,\mu_ n)\in {\mathfrak M}\), then the series \(\sum B_ n(\theta)/\log (n+1)\) is absolutely summable \((H,\mu_ n)\). Replacing \(h(t)\) by \(h_ 1(t)=t^{-1}\int^{t}_{0}h(u)du\), an analogous theorem is established.