Completeness in the set of wavelets

Abstract:

We study the completeness properties of the set of wavelets in $L^{2}(\mathbb {R})$. It is well-known that this set is not closed in the unit ball of $L^{2}(\mathbb {R})$. However, if one considers the metric inherited as a subspace (in the Fourier transform side) of $L^{2}(\mathbb {R},d\xi ) \cap L^{2}(\mathbb {R}_{*},{\frac {{d\xi }}{{|\xi |}}})$, we do obtain a complete metric space.