On unconditional and absolute convergence of wavelet type series

In this paper we consider wavelet type systems, i. e. systems of type
$$ \{\psi_{mn}(x)=2^{m/2}\psi(2^mx-n)\}, $$
where $\psi\in L^2(\mathbb R)$ such that $\operatorname{supp}\psi\Subset\mathbb R$. Let $E$ be a set of real numbers. We prove the equivalence of absolute and unconditional convergence almost everywhere on $E$ of the series
$$ \sum_{\substack{m\geq 0\\ n\in\mathbb Z}}a_{mn}\psi_{mn}(x) . $$