Kathy D. Merrill

Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ℝn

An abstract formulation of generalized multiresolution analyses is presented, and those GMRAs that come from multiwavelets are characterized. As an application of this abstract formulation, a constructive procedure is developed, which produces all wavelet sets in ℝnrelative to an integral expansive matrix.

Construction of Parseval wavelets from redundant filter systems

We consider wavelets in L2(Rd) which have generalized multiresolutions. This means that the initial resolution subspace V0 in L2(Rd) is not singly generated. As a result, the representation of the integer lattice Zd restricted to V0 has a nontrivial multiplicity function. We show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary-matrix-valued functions on a torus acting on a certain vector bundle. Specifically, we show how the wavelet functions on Rd can be constructed directly from the generalized wavelet filters.

The construction of wavelets from generalized conjugate mirror filters in L2(Rn)

Abstract The classical constructions of wavelets and scaling functions from conjugate mirror filters are extended to settings that lack multiresolution analyses. Using analogues of the classical filter conditions, generalized mirror filters are defined in the context of a generalized notion of multiresolution analysis. Scaling functions are constructed from these filters using an infinite matrix product. From these scaling functions, non-MRA wavelets are built, including one whose Fourier transform is infinitely differentiable on an arbitrarily large interval.

Smooth cocycles for an irrational rotation

Explicit examples of smooth cocycles not cohomologous to constants are constructed. Necessary and sufficient conditions on the irrational numberθ are given for the existence of such cocycles. It is shown that, depending onθ, the set ofCr cocycles whose skew-product is ergodic is either residual or empty.

Simple Wavelet Sets for Scalar Dilations in ℝ2

Wavelet sets for dilation by any scalar d > 1 in L2(ℝ2) are constructed that are finite unions of convex polygons. Such simple wavelet sets for dilation by 2 were widely conjectured to be impossible. The examples are built using the generalized scaling set technique of Baggett et al. [3]. Generalizations to other expansive dilations in L2(ℝ2) are discussed.

Probability measures on solenoids corresponding to fractal wavelets

The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen in (12) is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between solenoid measures associated with classical filters in R d and those associated with filters on inflated fractal sets. In particular, it is shown that the classical case produces atomic fiber measures, and as a result supports both suitably defined solenoid MSF wavelets and systems of imprimitivity for the corresponding wavelet representation of the generalized Baumslag-Solitar group. In contrast, the fiber measures for filters on inflated fractal spaces cannot be atomic, and thus can support neither MSF wavelets nor systems of imprimitivity.

Simple Wavelet Sets for Matrix Dilations in ℝ2

A partial answer is given to the question of which expansive integer matrix dilations in ℝ2 have wavelet sets that are finite unions of convex sets. New results are given supporting a conjecture that among matrices with determinant greater than 2, it is exactly matrices that have a power equal to a scalar that have such wavelet sets.

On the cohomological equivalence of a class of functions under an irrational rotation of bounded type

We prove that a linear combination of positive real powers of x, with integral 0 and equal values at 0 and 1 , is a coboundary for any irrational rotation of bounded type. We apply this result to establish the ergodicity of related compact and noncompact skew products.

Equivalence of cocycles under an irrational rotation

This paper describes a method for studying the equivalence relation among cocycles for an irrational rotation. A parameterized family of cocycles is presented, which meets the equivalence class of each piecewise absolutely continuous function whose derivative is L2. The difficulties in describing the equivalence among the elements of this family is shown to reduce to the analogous problem for describing equivalence among step functions, thereby relating this paper to the earlier work of Veech, Petersen, Merrill, and others.

Cocycles on the Circle. II

Γ is a countable dense subgroup of the circle T, which will be written additively (with addition modulo 2π). For each α in T the translation operator T α is defined by Tαf(x) = f(x + α). The aim of this note is to prove this result.