On the absolute matrix summability of Fourier series and some associated series (Q1331185)

Let \(f(t)\) be a periodic function with period \(2\pi\) and Lebesgue integrable in \((-\pi,\pi)\). Let \(f(t)\sim \sum^ \infty_ 0 A_ n(t)\) and let \(\sum^ \infty_ 1 B_ n(t)\) denote its conjugate series. The authors in this paper study the absolute matrix summability of \(\sum A_ n(x)\), \(\sum B_ n(x)\), \(\sum^ \alpha_ n A_ n(x)\), \(\sum^ \alpha_ n B_ n(x)\), \(0< \alpha<1\), and \(\sum{s_ n-s\over n}\), where \(s_ n\) denotes the \(n\)th partial sum of the series \(\sum^ \infty_ 0 A_ n(x)\). Their results generalize a number of well-known theorems including one due to the reviewer [Math. Scand. 21, 90-104 (1967; Zbl 0165.075)].