Multiresolution analysis on local fields and characterization of scaling functions

The concepts of multiresolution analysis (MRA) and wavelet can be generalized to a local field K of positive characteristic by using a prime element of such a field. An MRA is a sequence of closed subspaces of L-2 (K) satisfying certain properties. We show that it is enough to assume that the discrete translates of a single function in the core subspace of the MRA form a Riesz basis instead of an orthonormal basis and show how to construct an orthonormal basis from a Riesz basis. We also prove that the intersection triviality condition in the definition of MRA follows from the other conditions of an MRA. The union density condition also follows if we assume that the Fourier transform of the scaling function is continuous at 0. Finally we characterize the scaling functions associated with such an MRA.