Behaviour of an associated conjugate series of a Fourier series (Q1329248)

Let \(f(t)\) be a periodic function with period \(2 \pi\) and integrable \(L\) over \(( - \pi, \pi)\) and let \(f(t) \sim {1\over 2} a_ 0 + \sum^ \infty_{n = 1}\) \((a_ n \cos nt + b_ n \sin nt) = {1\over 2} a_ 0 + \sum^ \infty_{n = 1} A_ n (t)\). Then the series \(\sum^ \infty_{n = 1} (b_ n \cos nx - a_ n \sin nx) = \sum^ \infty_{n = 1} B_ n (x)\) is said to be the conjugate series of the Fourier series at \(t=x\). Let \(\psi (t) = {1\over 2} [f(x + t) - f(x - t)]\) and \(h(t)= {\psi (t) \over \log (a/t)}\) for \(a>\pi\). \textit{L. S. Bosanquet} and \textit{J. M. Hyslop} [Math. Z. 42, 489-512 (1937; Zbl 0016.21002)] proved that if \({| \psi (t) | \over t}\) is integrable in \((0, \pi)\) then the conjugate series of \(f(t)\) at \( = x\) is summable \(| C, 1 + \delta |\), \(\delta > 0\). \textit{R. Mohanty} and \textit{B. K. Ray} [Can. J. Math. 21, 535-551 (1969; Zbl 0179.369)] proved that if \(| h(t) |/t\) is integrable in \((0, \pi)\) then the series \(\sum^ \infty_{n = 1} [B_ n (x)/ \log (n + 1)]\) is summable \(| C,1 + \delta |\) for \(\delta > 0\). The authors are proving a general result from which the above due to Mohanty and Ray can be deduced as a particular case. They prove that ``If \(| h(t) |/t\) is integrable in \((0,\pi)\) then the series \(\sum^ \infty_{n = 1} [B_ n (x)/ \log (n + 1)]\) is summable \(| C, 1 + \delta, \beta |\) for all real \(\beta\) and \(\delta > 0\).