Multiresolution Approximations and Wavelet Bases of Weighted L ^p Spaces

Abstract

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 < ∞.