On the convergence of multiplicatively orthogonal series

Abstract:

G. Alexits and A. Sharma have recently shown that if $\{ {\varphi _n}\} _{n = 1}^\infty$ is a uniformly bounded multiplicatively orthogonal system on a finite measure space and if $\{ {c_n}\} _{n = 1}^\infty$ is a sequence of real numbers with $\sum _{n = 1}^\infty c_n^2 < \infty$, then the partial sums $\sum _{k = 1}^n{c_{k\varphi k}}$ converge almost everywhere. We give here a simple proof of this result.