On the linear independence of finite wavelet systems generated by Schwartz functions and functions with faster-than-exponential decay

One of the motivations for stating HRT conjecture on the linear independence of finite Gabor systems was the fact that there are linearly dependent Finite Wavelet Systems (FWS). This paper proves the linear independence of every FWS generated by a nonzero function with faster-than-exponential decay, provided the support of that function is not compact. It also proves the linear independence of every three-point FWS generated by a nonzero Schwartz function, and with any number of points if the FWS is generated by a nonzero Schwartz function whose Fourier transform is ultimately decreasing. In addition, we prove new results on the order of regularity of the solutions to the two-scale difference equation and the equation related to symmetric Bernoulli convolutions.