The sum 1/(n w(n)) converging is necessary and sufficient for an increasing w(n) to be an a.e. unconditional convergence Weyl multiplier for arbitrary wavelet-type systems, and log n is optimal for rearranged systems.
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Quantitative estimates for the absolute convergence of wavelet-type series
The sum 1/(n w(n)) converging is necessary and sufficient for an increasing w(n) to be an a.e. unconditional convergence Weyl multiplier for arbitrary wavelet-type systems, and log n is optimal for rearranged systems.