arXiv subjects

Judith A. Packer

Publications and source records attributed to Judith A. Packer.

14 recordsLinked to original sources

arXiv2022

Spectral Triples for Noncommutative Solenoids and a Wiener's lemma

In this paper we construct odd finitely summable spectral triples based on length functions of bounded doubling on noncommutative solenoids. Our spectral triples induce a Leibniz Lip-norm on the state spaces of the noncommutative solenoids, giving them the structure of Leibniz quantum compact metric spaces. By applying methods of R. Floricel and A. Ghorbanpour, we also show that our odd spectral triples on noncommutative solenoids can be considered as direct limits of spectral triples on rotation algebras. In the final section we prove a noncommutative Wiener's lemma and show that our odd spectral triples can be defined to have an associated smooth dense subalgebra which is stable under the holomorphic functional calculus, thus answering a question of B. Long and W. Wu. The construction of the smooth subalgebra also extends to the case of nilpotent discrete groups.

math.OA↗

arXiv2020

Decomposing the wavelet representation for shifts by wallpaper groups

The wavelet group and wavelet representation associated with shifts coming from a two dimensional crystal symmetry group $Γ$ and dilations by powers of 3, are defined and studied. The main result is an explicit decomposition of the $3Γ-$wavelet representation into irreducible representations of the wavelet group.

math.FA↗

arXiv2019

Yang-Mills connections on quantum Heisenberg manifolds

We investigate critical points and minimizers of the Yang-Mills functional YM on quantum Heisenberg manifolds $D^c_{μν}$, where the Yang-Mills functional is defined on the set of all compatible linear connections on finitely generated projective modules over the QHMs. A compatible linear connection which is both a critical point and minimizer of YM is called a Yang-Mills connection. In this paper, we investigate Yang-Mills connections with constant curvature. We are interested in Yang-Mills connections on the following classes of modules over the QHMs: (i) Abadie's module $Ξ$ of trace $2μ$ and its submodules; (ii) modules $Ξ^\prime$ of trace $2ν$; (iii) tensor product modules of the form $P E^c_{μν}\otimes Ξ$, where $E^c_{μν}$ is Morita equivalent to $D^c_{μν}$ and $P$ is a projection in $E^c_{μν}$. We present a characterization of critical points and minimizers of YM, and provide a class of new Yang-Mills connections with constant curvature via concrete examples. In particular, we show that every Yang-Mills connection $\nabla$ on $Ξ$ over $D^c_{μν}$ with constant curvature should have a certain form of the curvature such that $Θ_\nabla(X,Y)=Θ_\nabla(X,Z)=0$ and $Θ(Y,Z)=\frac{πi}μ Id_E$. Also we show that these Yang-Mills connections with constant curvature do not provide global minima but only local minima, and give two other examples which show that the critical points and minimizers of YM depend crucially on the geometric structure of the QHMs and of the projective modules over them. Furthermore, we construct the Grassmannian connection on the projective modules $Ξ^\prime$ with trace $2ν$ over the QHMs and compute its corresponding curvature, and we construct a tensor product connections on $P E^c_{μν}\otimes Ξ$ whose coupling constant is $2ν$ and characterize the critical points of YM for this projective module.

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arXiv2018

Generalized gauge actions on $k$-graph $C^*$-algebras: KMS states and Hausdorff structure

For a finite, strongly connected $k$-graph $Λ$, an Huef, Laca, Raeburn and Sims studied the KMS states associated to the preferred dynamics of the $k$-graph $C^*$-algebra $C^*(Λ)$. They found that these KMS states are determined by the periodicity of $Λ$ and a certain Borel probability measure $M$ on the infinite path space $Λ^\infty$ of $Λ$. Here we consider different dynamics on $C^*(Λ)$, which arise from a functor $y: Λ\to \mathbb{R}_+$ and were first proposed by McNamara in his thesis. We show that the KMS states associated to McNamara's dynamics are again parametrized by the periodicity group of $Λ$ and a family of Borel probability measures on the infinite path space. Indeed, these measures also arise as Hausdorff measures on $Λ^\infty$, and the associated Hausdorff dimension is intimately linked to the inverse temperatures at which KMS states exist. Our construction of the metrics underlying the Hausdorff structure uses the functors $y: Λ\to \mathbb{R}_+$; the stationary $k$-Bratteli diagram associated to $Λ$; and the concept of exponentially self-similar weights on Bratteli diagrams.

math.OA↗

arXiv2010

Probability measures on solenoids corresponding to fractal wavelets

The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen 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 $\mathbb R^d$ 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.

math.CA↗

arXiv2009

Classification of Generalized Multiresolution Analyses

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 $L^2 (\mathbb R^d)$ as well as to previously studied abstract examples.

math.CA↗

arXiv2008

Generalized low-pass filters and multiresolution analyses

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.

math.CA↗

arXiv2008

Direct limits, multiresolution analyses, and wavelets

A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on $L^2(\R^n)$, the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids.

math.FA↗

arXiv2007

Generalized multiresolution analyses with given multiplicity functions

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 $L^2(\mathbb R^n)$, 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 $m$.

math.FA↗

arXiv2005

Projective multiresolution analyses for dilations in higher dimensions

We continue the study of projective module wavelet frames corresponding to diagonal dilation matrices on $\mathbb R^n$ with integer entries, focusing on the construction of a projective multi-resolution analysis corresponding to dilations whose domains are finitely generated projective modules over continuous complex-valued functions on $n$-tori, $n\geq 3$. We are able to generalize some of these results to dilation matrices that are conjugates of integral diagonal dilation matrices by elements of $SL(n,\mathbb Z).$ We follow the method proposed by the author and M. Rieffel, and are able to come up with examples of non-free projective module wavelet frames which can be described via this construction. As an application of our results, in the case $n=3,$ when the dilation matrix is a constant multiple of the identity, we embed every finitely generated module as an initial module.

math.FA↗

arXiv2004

Applications of the work of Stone and von Neumann to wavelets

This survey paper examines the work of J. von Neumann and M.H. Stone as it relates to the abstract theory of wavelets. In particular, we discuss the direct integral theory of von Neumann and how it can be applied to representations of certain discrete groups to study the existence of normalized tight frames in the setting of Gabor systems and wavelets, via the use of group representations and von Neumann algebras. Then the extension of Stone's theorem due to M. Naimark, W. Ambrose and R. Godement is reviewed, and its relationship to the multiresolution analyses of S. Mallat and Y. Meyer and the generalized multiresolution analyses of L. Baggett, H. Medina, and K. Merrill. Finally, the paper ends by discussing some recent work due to the author, Baggett, P. Jorgensen and Merrill, and its relationship to operator theory.

math.FA↗

arXiv2004

Projective multiresolution analyses for $L^2(R^2)$

We define the notion of "projective" multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra $C(\btn)$ of continuous complex-valued functions on an $n$-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyses, including the frames which they provide for $L^2(\brn)$. Then we show how to construct examples for the case of any diagonal $2 \times 2$ dilation matrix with integer entries, with initial module specified to be any fixed finitely generated projective $C(\mathbb T^2)$-module. We compute the isomorphism classes of the corresponding wavelet modules.

math.FA↗

arXiv2002

Wavelet filter functions, the matrix completion problem, and projective modules over $C(\mathbb T^n)$

We discuss how one can use certain filters from signal processing to describe isomorphisms between certain projective $C(\mathbb T^n)$-modules. Conversely, we show how cancellation properties for finitely generated projective modules over $C(\mathbb T^n)$ can often be used to prove the existence of continuous high pass filters, of the kind needed for multivariate wavelets, corresponding to a given continuous low-pass filter. However, we also give an example of a continuous low-pass filter for which it is impossible to find corresponding continuous high-pass filters. In this way we give another approach to the solution of the matrix completion problem for filters of the kind arising in wavelet theory.

math.FA↗

arXiv2001

A Direct Integral Decomposition of the Wavelet Representation

In this paper we use the concept of wavelet sets as introduced by X. Dai and D. Larson, to decompose the wavelet representation of the discrete group associated to an arbitrary $n \times n$ integer dilation matrix as a direct integral of irreducible monomial representations. In so doing we generalize a result of F. Martin and A. Valette in which they show that the wavelet representation is weakly equivalent to the regular representation for the Baumslag-Solitar groups.

math.FA↗