WANDERING VECTORS FOR UNITARY SYSTEMS AND ORTHOGONAL WAVELETS - INTRODUCTION

We investigate topological and structural properties of the set W(U) o f all complete wandering vectors for a system U of unitary operators a cting on a Hilbert space. The special case of greatest interest is the system [D,T] of dilation (by 2) and translation (by 1) unitary operat ors acting on L2(R), for which the complete wandering vectors are prec isely the orthogonal dyadic wavelets. The method we use is to paramete rize W(U) in terms of a fixed vector psi and the set of all unitary op erators which locally commute with U at psi. An analysis of the struct ure of this local commutant yields new information about W(U). The com mutant of a unitary system can be abelian and yet the local commutant of it at a complete wandering vector can contain noncommutative von Ne umann algebras as subsets. This is the case for [D,T]. The unitary gro up of a certain non-commutative von Neumann algebra can be used to par ameterize a connected class of wavelets generalizing those of Meyer wi th compactly supported Fourier transform.