Multiresolution Approximations and Wavelet Bases of WeightedLpSpaces

We study boundedness and convergence on L ^p (R ⁿ ,d) 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 < .