On the Density Condition of a Multiresolution Analysis in Lebesgue Spaces

Abstract

We are interested in the problem of when the density condition in a multiresolution analysis defined in Lp(Rn), 1≤p≤∞, holds. Indeed, if 2≤p<∞, we obtain sufficient conditions on the generators of a multiresolution analysis in order to the density condition is satisfied. We emphasis on the requirement of the Fourier transform in a neighborhood of the origin. This involves the notion of density point. When 1≤p≤2, the obtained condition is necessary. Moreover, we study the same problem when a multiresolution analysis is defined in the subspace of L∞(Rn) of the set of all continuous functions vanishing at infinite.