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

Adam Paszkiewicz

paper

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

arXiv:math/0507568

Abstract

We characterize sequences of numbers $(a_{n})$ such that $\sum_{n \geq 1} a_{n} Φ_{n}$ 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 $h_{B} (t) = - lo g_{3} (β - α)$ for $t \in (α, β]$ , $[α, β] \cap B = {α, β} .$

44 pages