The Theory of Wavelets with Composite

Summary. A wavelet with composite dilations is a function generating an orthonor-mal basis or a Parseval frame for L2(Rn) under the action of lattice translations and dilations by products of elements drawn from non-commuting sets of matrices A and B. Typically, the members of B are matrices whose eigenvalues have magnitude one, while the members of A are matrices expanding on a proper subspace of Rn. The theory of these systems generalizes the classical theory of wavelets and pro-vides a simple and exible framework for the construction of orthonormal bases and related systems that exhibit a number of geometric features of great potential in applications. For example, composite wavelets have the ability to produce \long and narrow " window functions, with various orientations, well-suited to applications in image processing. Dedicated to John J. Benedetto. We assume familiarity with the basic properties of separable Hilbert spaces. L2(Rn), the space of all square integrable functions on Rn, and L2(Tn), th