Wandering vectors for unitary systems and orthogonal wavelets

We investigate topological and structural properties of the set W(U) of all complete wandering vectors for a system U of unitary operators acting on a Hilbert space. The special case of greatest interest is the system D, T of dilation (by 2) and translation (by 1) unitary operators acting on L2(), for which the complete wandering vectors are precisely the orthogonal dyadic wavelets. The method we use is to parameterize W(U) in terms of a fixed vector and the set of all unitary operators which locally commute with U at . An analysis of the structure of this local commutant yields new information about W(U). The commutant of a unitary system can be abelian and yet the local commutant of it at a complete wandering vector can contain non-commutative von Neumann algebras as subsets. This is the case for D, T. The unitary group of a certain non-commutative von Neumann algebra can be used to parameterize a connected class of wavelets generalizing those of Meyer with compactly supported Fourier transform.

Wandering vectors for unitary systems and orthogonal wavelets Academic Article