Absolute Riesz summability factors of Fourier series
Let \(f\in L\) be a 2\(\pi\)-periodic function and let \(f(t)\sim \sum^{\infty}_{n=1}(a_ n\quad \cos nt+b_ n\quad \sin nt)\equiv \sum^{\infty}_{n=1}A_ n(t)\). Set \(\phi_ 0(t)=\phi(t)=((f(x+t)+f(x-t))/2\) and \(\phi_{\beta}(t)=\beta t^{- \beta}\int^{t}_{0}(t-u)^{\beta -1}\phi(u)du, \beta>0\). In this paper the author proves the following Theorem: Let \(0\leq \beta<1\), \(\delta\geq 0\), \(0<s+\beta<\delta +\beta<1\) and \(\alpha =(\delta - s)/(\delta +\beta)\) then \(t^{-\delta}\phi_{\beta}(t)\in BV[0,\pi]\) implies that \(\sum^{\infty}_{n=1}n^ sA_ n(x)\) is summable \(| R,\quad \exp \omega^{\alpha},\quad r|,\) with \(r>\delta +\beta\). Using this theorem, the author deduces known results due to \textit{K. Matsumoto} [Tôhoku Math. J., II. Ser. 8, 205-222 (1956; Zbl 0075.053)] and \textit{G. D. Dikshit} [Kyungpook Math. J. 17, 195-201 (1977; Zbl 0382.42005), J. Aust. Math. Soc., Ser. A 19, 97-102 (1975; Zbl 0293.42005), Indian J. Math. 18, 117-124 (1976; Zbl 0373.42006)].
scientific article; zbMATH DE number 3855694
scientific article