Kathy Merrill

An analogue of Bratteli-Jorgensen loop group actions for GMRA's

Several years ago, O. Bratelli and P. Jorgensen developed the concept of m-systems of filters for... more Several years ago, O. Bratelli and P. Jorgensen developed the concept of m-systems of filters for dilation by a positive integer N>1 on L^2(R). They constructed a loop group action on m-systems. By work of Mallat and Meyer, these m-systems are important in constructing multi-resolution analyses and wavelets associated to dilation by N and translation by Z on L^2(R). In

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Fractal wavelets of Dutkay-Jorgensen type for the Sierpinski gasket space

Frames and Operator Theory in Analysis and Signal Processing, 2008

Abstract. Several years ago, D. Dutkay and P. Jorgensen developed the concept of wavelets defined... more

$Research paper thumbnail of Probability measures on solenoids corresponding to fractal wavelets$

Probability measures on solenoids corresponding to fractal wavelets

Trans. Amer. Math. Soc., 2012

The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen in is anal... more The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen in 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.

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Representations of the Mautner group and cocycles of an irrational rotation

The Michigan Mathematical Journal, 1986

Representations, Wavelets, and Frames

... Jonas D&amp;#x27;Andrea, Gail Ratcliﬀ, Dorin Dutkay, Eberhard Kaniuth (partially blocked)... more

Generalized low-pass filters and multiresolution analyses

We study generalized filters that are associated to multiplicity functions and homomorphisms of t... more We study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group. These notions are based on the structure of generalized multiresolution analyses. We investigate when the Ruelle operator corresponding to such a filter is a pure isometry, and then use that characterization to study the problem of when a collection of closed subspaces, which satisfies all the conditions of a GMRA except the trivial intersection condition, must in fact have a trivial intersection. In this context, we obtain a generalization of a theorem of Bownik and Rzeszotnik.

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Equivalence of cocycles under an irrational rotation

Proceedings of the American Mathematical Society, 1988

On Functions that are Trivial Cocycles for a Set of Irrationals

by Herbert Medina and Kathy Merrill

Proceedings of The American Mathematical Society, 1988

Abstract. Two results are obtained about the topological size of the set of irrationals for which... more Abstract. Two results are obtained about the topological size of the set of irrationals for which a given function is a trivial cocycle. An example,of a continuous function which is a coboundary,with non-L, cobounding function is constructed. A function v :R=Z!R is called an (additive) coboundary for an irrational if

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

Proceedings of the American Mathematical Society, 1991

... 2^(5&amp;quot;^&amp;#x27;« «i &amp;gt; Mx ln(Nk) for some constant Mx . We see by... more

Continued Fractions and Series

Journal of Number Theory, 1995

Explicit formulae are given relating continued fractions with almost periodic or almost symmetric... more Explicit formulae are given relating continued fractions with almost periodic or almost symmetric patterns in their partial quotients, and infinite series whose terms satisfy certain recurrence relations. For quadratic irrationals, the continued fraction expansions for successive approximations given by Newton′s method are described.

Construction of Parseval wavelets from redundant filter systems

Journal of Mathematical Physics, 2005

We consider wavelets in L 2 (R d ) which have generalized multiresolutions. This means that the i... more We consider wavelets in L 2 (R d ) which have generalized multiresolutions. This means that the initial resolution subspace V 0 in L 2 (R d ) is not singly generated. As a result, the representation of the integer lattice Z d restricted to V 0 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 R d can be constructed directly from the generalized wavelet filters.

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Generalized filters, the low-pass condition, and connections to multiresolution analyses

Journal of Functional Analysis, 2009

We study generalized filters that are associated to multiplicity functions and homomorphisms of t... more We study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group. These notions are based on the structure of generalized multiresolution analyses. We investigate when the Ruelle operator corresponding to such a filter is a pure isometry, and then use that characterization to study the problem of when a collection of closed subspaces, which satisfies all the conditions of a GMRA except the trivial intersection condition, must in fact have a trivial intersection. In this context, we obtain a generalization of a theorem of Bownik and Rzeszotnik.

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Classification of generalized multiresolution analyses

Journal of Functional Analysis, 2010

We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on ... more We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions m and matrix-valued filter functions H. Given a natural number valued function m and a system of functions encoded in a matrix H satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function m and filter system H. An equivalence relation on GMRAs is defined and described in terms of their associated pairs (m, H). This classification system is applied to classical examples in L 2 (R d ) as well as to previously studied abstract examples.

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Generalized Multiresolution Analyses with Given Multiplicity Functions

Journal of Fourier Analysis and Applications, 2009

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space H t... more Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space H that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space H is L 2 (R n ), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.

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

by Herbert Medina and Kathy Merrill

Journal of Fourier Analysis and Applications, 1999

An abstract formulation of generalized multiresolution analyses is presented, and those GMRAs tha... more 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.

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Smooth cocycles for an irrational rotation

Israel Journal of Mathematics, 1992

Explicit examples of smooth cocycles not cohomologous to constants are constructed. Necessary and... more 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 ofC r cocycles whose skew-product is ergodic is either residual or empty.

The construction of wavelets from generalized conjugate mirror filters in

Applied and Computational Harmonic Analysis, 2002

The classical constructions of wavelets and scaling functions from conjugate mirror filters are e... more 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.

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Generalized Multiresolution Analyses, And A Construction Procedure For All Wavelet Sets In

by Herbert Medina and Kathy Merrill

An abstract formulation t~'generalized multiresolution analyses is presented, and those GMRAs tha... more An abstract formulation t~'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 Nn relative to an integral expansive matrix.

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On Functions That Are Trivial Cocycles for a Set of Irrationals. II

by Herbert Medina and Kathy Merrill

Two results are obtained about the topological size of the set of irrationals for which a given f... more

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Cohomology of polynomials under an irrational rotation

Proceedings of the American …, 1998

Abstract. A new description of cohomology of functions under an irrational rotation is given in t... more

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