We study boundedness and convergence on L p(R n,dμ) of the projection operators P j given by MRA structures with non-necessarily compactly supported scaling function. As a consequence, we prove that if w is a locally integrable function such that w -(1/p–1)(x) (1+|x|)-N is integrable for some N > 0, then the Muckenhoupt A p condition is necessary and sufficient for the associated wavelet system to be an unconditional basis for the weighted space L p(R n,w(x) dx), 1 < p < ∞.
