Dilates of shift-invariant spaces on local fields

Let K be a local field of positive characteristic. We prove that if the space V of negative dilates of a Parseval wavelet of L2(K) has dimension function finite on a set of positive measure, then the intersection of the dilates of V is trivial. We also construct an example of a frame wavelet of L2(K) whose space of negative dilates is all of L2(K). The frame wavelet can be chosen to have frame bounds arbitrarily close to 1 and it has a dual frame wavelet.