Quantitative estimates for the absolute convergence of wavelet-type series

Abstract

We establish new quantitative estimates for general systems of functions with wavelet-type dyadic structure. These estimates are applied to obtain the optimal growth of various types of Weyl multipliers for certain wavelet-type systems. Some of our results are sufficiently general to allow the orthogonality assumption to be removed. In particular, as a consequence of these estimates we show that the condition \begin{equation*} \sum_{n=1}^\infty\frac{1}{nw(n)}<\infty \end{equation*} is necessary and sufficient for an increasing sequence $w(n)$ to be an almost everywhere unconditional convergence Weyl multiplier for an arbitrary wavelet-type system. We also prove that $\log n$ is an almost everywhere convergence Weyl multiplier for any rearranged wavelet-type system, and that this bound is optimal.