Almost everywhere convergence of orthogonal series (Q1087770)

We say that a function \(\delta\) (x) is a control function for an almost everywhere convergence of \(f_ n(x)\) to f(x) on [0,1], if for every \(\epsilon >0\) there exists an integer n(\(\epsilon)\) such that \(| f_ n(x)-f(x)| <\epsilon \delta (x)\) everywhere for \(n\geq n(\epsilon)\). [Cf. \textit{K. Yoneda}, Math. Jap. 20, 101-105 (1975; Zbl 0314.28005)]. Let \(\{\phi_ n(x)\}\) be an orthogonal system on [0,1], \(\sum^{\infty}_{n=0}c^ 2_ n<\infty\) and \(\sigma_ n^{\alpha}(x)\) the (C,\(\alpha)\)-mean of \(\sum c_ n\phi_ n\). The following is the main result of the paper: suppose that \(k>0\), \(\gamma =\min (1,k/2)\) and \(\{\beta >1-\alpha /k\), \(0<\alpha \leq \gamma \}\) or \(\{\gamma <\alpha <\infty,\beta >1-\gamma /k\}\). If \(\sigma_ n^{\beta}(x)\) converges to f(x) a.e. on [0,1] and it has an \(L^ p\)- integrable control function for some \(p>0\), then \[ \lim_{n\to \infty}(1/A_ n^{\alpha})\sum^{n}_{\nu =0}A^{\alpha -1}_{n- \nu}| \sigma_{\nu}^{\beta -1}(x)-f(x)|^ k=0 \] a.e. on [0,1], and the above convergence has an \(L^ q\)-integrable control function, where \(q=\min (2,p)\).