On a Littlewood-Paley Identity and Characterization of Wavelets

A W-family is a family of 2s − 1 functions in L2(Rs), s ≥ 1, whose binary dilations and integer translates form a Riesz basis of L2(Rs) such that the dual Riesz basis is also generated by 2s − 1 functions by the same operations. An objective of this paper is to show that any W-family satisfies an identity of Littlewood-Paley type, and conversely, this identity can be used to partially characterize a W-family. In addition, Littlewood-Paley identities for dyadic wavelets and frames along with their duals are also establised. In particular, such an identity is used to characterize dual dyadic wavelets for perfect reconstruction form information of integral wavelet transforms on dyadic resolutions and continous-space domains. It is also shown that dual dyadic wavelets are not unique in general.