Unitary systems and wavelet sets

A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. We will describe an operator-interpolation approach to wavelet theory using the local commutant of a unitary system. This is an application of the theory of operator algebras to wavelet theory. The concrete applications to wavelet theory include results obtained using specially constructed families of wavelet sets. The main section of this paper is section 5, in which we introduce the interpolation map a induced by a pair of wavelet sets, and give an exposition of its properties and its utility in constructing new wavelets from old. The earlier sections build up to this, establishing terminology and giving examples. The main theoretical result is the Coefficient Criterion, which is described in Section 5.2.2, and which gives a matrix valued function criterion specificing precisely when a function with frequency support contained in the union of an interpolation family of wavelet sets is in fact a wavelet. This can be used to derive Meyer's famous class of wavelets using an interpolation pair of Shannon-type wavelet sets as a starting point. Section 5.3 contains a new result on interpolation pairs of wavelet sets: a proof that every pair of sets in the generalized Journe family of wavelet sets is an interpolation pair. We will discuss some results that are due to this speaker and his former and current students. And we finish in section 6 with a discussion of some open problems on wavelets and frame-wavelets.