On Weyl factors for certain classes of orthonormed systems (Q1842471)

Let \(\{f_ n(x)\}\) be an orthonormed system of functions on \([0, 1]\) with \(f_ n\in L_ p[0, 1]\) for certain \(2< p< \infty\). Let \(\Lambda= \{\lambda_ k\}\) be a non-decreasing sequence of real numbers. System \(\{f_ k(x)\}\) is termed \(S(p, \Lambda)\)-system if an arbitrary polynomial \(P_ n(x)= \sum^ n_{k= 1} a_ k f_ k(x)\) satisfies the inequality \(\| P_ n\|_ p\leq \lambda_ n\| P_ n\|_ 2\). For a series (*) \(\sum^ \infty_{k= 1} a_ k f_ k(x)\) let be \(S^*(x)= \sup_{1\leq k< \infty} \left|\sum^ k_{j= 1} a_ j f_ j(x)\right|\). Theorem. Let \(\{f_ k(x)\}\) be an \(S(p, \Lambda)\)-system \((2< p< \infty)\). If \[ \Omega= \sum^ \infty_{k= 1} a^ 2_ k\ln^{2+ \varepsilon}(e+ \lambda_ k)< \infty \] for some \(\varepsilon> 0\), then the series (*) converges almost everywhere on \([0,1]\), and \(\| S^*\|\leq O(\Omega^{1/2})\) holds. This is valid for the rearranged series \(\sum^ \infty_{k= 1} a_{\sigma(k)} f_{\sigma(k)}(x)\) also.