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How linearly independent are orthonormal wavelet systems (Daubechies wavelets)?

I am mostly interested in compactly supported wavelet systems, and in particular in Daubechies wavelets, but other cases and also homogeneous wavelet systems would be of interest. … I am interested in this question, since most of the sources regarding the characterization of Besov spaces via wavelets (for instance Theorem 1.20 in Triebel's book "Function spaces and wavelets on domains …