On the absolute summability of associated Fourier series (Q1074043)

Let f(t) be Lebesgue integrable and periodic with period \(2\pi\). Let \(f(x)=\sum^{\infty}_{0}A_ n(x)\), \(s_ n=\sum^{n}_{k=0}A_ k(x)\), \(\phi (t)=1/2\{f(x+t)+f(x-t)-2s\}\), \(\phi_ 1(t)=1/t\int^{t}_{0}\phi (u)du\). The main result of the author: Theorem. Let x(\(\nu)\) satisfy the following conditions: (a) for \(\nu\geq 1/\pi\), x(\(\nu)\) is positive and continuous. Also x(\(\nu)\) is of bounded variation in any finite interval in \(\nu\geq 1/\pi\); (b) there is a non- negative constant \(\lambda <1\) such that for sufficiently large \(\nu\), \(\nu^{\lambda}x(\nu)\) is non-decreasing and \(\nu^{-\lambda}x(\nu)\) is non-increasing. Suppose that (i) \(x(1/t)\phi_ 1(t)\in BV(0,\pi)\), (ii) \(\{x(1/t)\phi_ 1(t)\}/t\in L(0,\pi)\); then the series (*) \(\sum^{\infty}(s_ n-s)x(n)/n\) is summable \(| C,1|\). If, in addition, x(\(\nu)\) is monotonic nondecreasing in [1/\(\pi\),\(\infty)\), then the series (*) is summable \(| C,\alpha |\), \(\alpha >\lambda\). \textit{M. K. Nayak} [Math. Proc. Camb. Philos. Soc. 70, 421-433 (1971; Zbl 0259.42014)] has investigated the absolute summability of the associated Fourier series.