Multiresolution approximations and wavelet bases of weighted \(L^p\) spaces
In this paper necessary and sufficient conditions on the Borel measure \( \mu \) of \(\mathbb R^n\) are obtained for which there is a mean and almost everywhere convergence of multiresolution approximations of \( L^p(d\mu ) \) functions if the scaling function is assumed to be continuous and absolutely bounded by an \( L^1 \) radial decreasing function. The approach relies on a weak positivity property of the kernels of the projection operators. Further, necessary and sufficient conditions on the Borel measure \( \mu \) for the associated wavelet systems to be unconditional bases for the space \( L^p(d\mu ) \) are studied.
weighted Lebesgue spaces, unconditional basis
scientific article; zbMATH DE number 2102103