On the convergence of series of weakly multiplicative systems of functions

V. F. Gaposhkin

Sbornik. Mathematics, Tome 18 (1972) no. 3, pp. 361-372

Résumé

A system of measurable functions $\{\varphi_k\}$ defined on a measurable space is called weakly multiplicative if it satisfies the relations $$ \int_X\varphi_{k_1}\varphi_{k_2}\dots\varphi_{k_p}\,d\mu=0\quad(\forall p\geqslant2,\ k_1\dots). $$ In this paper the convergence in the metric of $L_p$ and a.e. is investigated for series of weakly multiplicative system of functions. One of the results is: {\it If $\{\varphi_k\}$ is weakly multiplicative and $\sup_k\|\varphi_k\|_p\leqslant M$ for some $p>2,$ then any series $\sum c_k\varphi_k$ with coefficients in $l_2$ converges unconditionally a.e. and in $L_p$}. For $p=2n$, instead of weak multiplicativity it is sufficient to require the condition $\int_X\varphi_{k_1}\dots\varphi_{k_{2n}}\,d\mu=0$ ($\forall k_1\dots$). Bibliography: 13 titles.