Über die Mittel von orthogonalen Funktionen (Q797085)

Let \(\Omega\) denote the set of all systems of real functions \(\phi =\{\phi_ n(x)\}\) orthonormal over (0,1). Let \(a=\{a_ n\}\) be a given sequence of real numbers, and let \(\lambda =\{\lambda_ n\}\) be a non- decreasing sequence of positive numbers with \(\lambda_ n\to \infty\) as \(n\to \infty\). We define \[ \| a;\lambda \| =\sup_{\phi \in \Omega}\{\int^{1}_{0}\sup_{n}(1/\lambda_ n\sum^{n}_{k=1}a_ k\phi_ k(x))^ 2dx\}^{{1\over2}}. \] It is proved that if \(\| a;\lambda \|<\infty\) then, for every \(\phi \in \Omega\), we have \((1)\quad(1/\lambda_ n)\sum^{n}_{k=1}a_ k\phi_ k(x)\to 0\) almost everywhere. If \(\| a;\lambda \| =\infty\) then there is some \(\phi \in \Omega\) such that \(\lim \sup_{n\to \infty}(1/\lambda_ n)| \sum^{n}_{k=1}a_ k\phi_ k(x)| =\infty\) almost everywhere. Similar results but with \(\lambda_ n\) replaced by 1 and with (1) replaced by the convergence of \(\sum^{\infty}_{k=1}a_ k\phi_ k(x)\) had previously been given by the author [Acta Sci. Math. 25, 219-232 (1964; Zbl 0135.274)]. Other results belonging to the same field of ideas are also obtained.