Absolute Riesz summability of conjugate series with factors (Q1330830)

Let \(f\) be a Lebesgue integrable \(2\pi\)-periodic function and \(\sum_ n A_ n(x)\) be its Fourier series. Assuming that under certain restrictions on \(k\), \((f(x+ t)+ f(x- t))(\log(2\pi/t))^ k\) is of bounded variation in \((0,\pi)\), \textit{G. D. Dikshit} and \textit{C. S. Rees} [Proc. Am. Math. Soc. 82, 231-238 (1981; Zbl 0459.42011)] have proved a result for absolute summability of \(\sum_ n A_ n(x)(\log n)^ m\) by a certain Riesz method. The authors prove a corresponding analogue for the conjugate series.