On complete characterization of coefficients of a.e. converging orthogonal series

Abstract

We characterize sequences of numbers $(a_n)$ such that $\sum_{n\geq 1} a_nΦ_n$ converges a.e. for any orthonormal system $(Φ_n)$ in any $L_2$-space. In our criterion, we use the set $B =\{\sum_{m\geq n} |a_m|^2; n\geq 1\}$ and its information function $$h_B(t) = -\log_3(β-α)$$ for $t\in (α, β]$, $[α, β]\cap B =\{α, β\}.$