Abstract:

The Generalized frame multiresolution analysis of an abstract Hilbert space (GFMRA) is an increasing chain of closed susbpaces of a Hilbert space generated by the action of a cyclic unitary group refered to as the Dilation group, on a particular element of the GFMRA, called Core subspace. The core subspace has a frame generated by the action of another abelian unitary group, refered to as Translation group, on a countable set of vectors in the core subspace. This particular set of vectors is called set of frame multiscaling vectors. We prove that for every GFMRA a set of frame multiwavelet vectors associated with this GFMRA can be constructed. We will present the main ideas of two different constructions giving perhaps different sets of frame multiwavelet vectors. Also, we develop the generalization of the concept of Quadratic Mirror filters and we characterize all the sets of frame multiwavelet vectors associated with given GFMRA. Finaly, we show how ALL orthonormal wavelets of L^2(R) are associated with GFMRAs. Our techniques are based on characterizations of commutants of certain Von Neumann algebras. We correlate these characterizations with the so-called fiberization technique of Ron and Shen to derive our results, which generalize the first foundamental classical results of multiresolution theory developed by Mallat and Meyer.