Kathy Merrill - Academia.edu (original) (raw)
Papers by Kathy Merrill
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
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 Abstract. Several years ago, D. Dutkay and P. Jorgensen developed the concept of wavelets defined on a σ-finite fractal measure space, developed from an iterated affine system. They worked out in detail the wavelet and filter functions corresponding to the ordinary Cantor ...
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.
Representations of the Mautner group and cocycles of an irrational rotation
The Michigan Mathematical Journal, 1986
Representations, Wavelets, and Frames
... Jonas D'Andrea, Gail Ratcliff, Dorin Dutkay, Eberhard Kaniuth (partially blocked)... more ... Jonas D'Andrea, Gail Ratcliff, Dorin Dutkay, Eberhard Kaniuth (partially blocked), Kuzman Adzievski, Kevin Manley, Christian Roldan-Santos, John ... Kneeling, left to right: Marcin Bownik, Demetrio Labate, Judy Packer, Kathy Merrill, Karla Oty, Wannapa Ruangthanakorn, Fumiko ...
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.
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
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"^'« «i > Mx ln(Nk) for some constant Mx . We see by... more ... 2^(5"^'« «i > Mx ln(Nk) for some constant Mx . We see by Lemma 2 that {ern} does not satisfy the hypothesis of the Marcin-kiewicz Theorem. In the following theorem, we apply Marcinkiewicz using an appropriately modified sequence. ...
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.
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.
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.
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.
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.
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.
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.
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.
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.
Two results are obtained about the topological size of the set of irrationals for which a given f... more 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 1 cobounding function is constructed.
Proceedings of the American …, 1998
Abstract. A new description of cohomology of functions under an irrational rotation is given in t... more Abstract. A new description of cohomology of functions under an irrational rotation is given in terms of symmetry properties of the functions on subinter-vals of [0, 1]. This description yields a method for passing information about the cohomology classes for a given irrational to ...
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
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 Abstract. Several years ago, D. Dutkay and P. Jorgensen developed the concept of wavelets defined on a σ-finite fractal measure space, developed from an iterated affine system. They worked out in detail the wavelet and filter functions corresponding to the ordinary Cantor ...
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.
Representations of the Mautner group and cocycles of an irrational rotation
The Michigan Mathematical Journal, 1986
Representations, Wavelets, and Frames
... Jonas D'Andrea, Gail Ratcliff, Dorin Dutkay, Eberhard Kaniuth (partially blocked)... more ... Jonas D'Andrea, Gail Ratcliff, Dorin Dutkay, Eberhard Kaniuth (partially blocked), Kuzman Adzievski, Kevin Manley, Christian Roldan-Santos, John ... Kneeling, left to right: Marcin Bownik, Demetrio Labate, Judy Packer, Kathy Merrill, Karla Oty, Wannapa Ruangthanakorn, Fumiko ...
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.
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
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"^'« «i > Mx ln(Nk) for some constant Mx . We see by... more ... 2^(5"^'« «i > Mx ln(Nk) for some constant Mx . We see by Lemma 2 that {ern} does not satisfy the hypothesis of the Marcin-kiewicz Theorem. In the following theorem, we apply Marcinkiewicz using an appropriately modified sequence. ...
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.
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.
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.
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.
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.
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.
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.
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.
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.
Two results are obtained about the topological size of the set of irrationals for which a given f... more 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 1 cobounding function is constructed.
Proceedings of the American …, 1998
Abstract. A new description of cohomology of functions under an irrational rotation is given in t... more Abstract. A new description of cohomology of functions under an irrational rotation is given in terms of symmetry properties of the functions on subinter-vals of [0, 1]. This description yields a method for passing information about the cohomology classes for a given irrational to ...