We establish several results yielding linear independence of the affine system generated by $\psi$ in exchange for conditions on the space $V(\psi)$ of negative dilates. A typical assumption yielding linear independence is that the space $V(\psi)$ is shift-invariant. In particular, the affine system generated by a Parseval wavelet is linearly independent. As an illustration of our techniques, we give an alternative proof of the theorem of Linnell (see Proc. Amer. Math. Soc. 127 (1999), 3269–3277) on linear independence of Gabor systems.
"Linear independence of Parseval wavelets." Illinois J. Math. 54 (2) 771 - 785, Summer 2010. https://doi.org/10.1215/ijm/1318598681