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Double Projective Approximation and Projective Decomposition in the Construction of Wavelet Bases

The usual method for the construction of a wavelet basis in the Hilbert space L 2 (R) via a Multiresolution Analysis (MRA) does not necessarily succeed in a Banach space, although MRAs can be constructed easily, as e.g. L 1 (R) shows. We explain this phenomenon by considering sequences of projections rather than subspaces, and by studying which properties of projection operators can or can not be derived from properties of their images. We then show that from an MRA, a wavelet basis for a certain subspace can be constructed.