arXiv subjects

Ursula Molter

Publications and source records attributed to Ursula Molter.

At least 19 recordsLinked to original sources

arXiv2026

Dynamical Sampling: A Survey

Dynamical sampling refers to a class of problems in which space-time samples are taken from a signal evolving under an underlying dynamical system. The goal is to use these samples to recover relevant information about the system, such as the initial state, the evolution operator, or the sources and sinks driving the dynamics. These problems are tightly connected to frame theory, operator theory, functional analysis, and other foundational areas of mathematics; they also give rise to new theoretical questions and have applications across engineering and the sciences. This survey provides an overview of the theoretical underpinnings of dynamical sampling, summarizes recent results, and outlines directions for future work, including open problems and conjectures.

math.FA↗

arXiv2025

Two-Scale Frostman Measures

We establish a unified Frostman-type framework connecting the classical Hausdorff dimension with the family of intermediate dimensions $\dim_θ$ recently introduced by Falconer, Fraser and Kempton. We define a new geometric quantity $\mathcal{D}(E)$ and prove that, under mild assumptions, there exists a family of measures $\{μ_δ\}$ supported on $E$ satisfying two simultaneous decay conditions, corresponding to the Hausdorff and intermediate Frostman inequalities. Such $(δ, s, t)$-Frostman measures allow for a two-scale characterization of the dimension of $E$.

math.CA↗

arXiv2025

Critical values for Intermediate and Box dimension of projections and other images

Given a compact set $E\subset\mathbb{R}^d$ we investigate for which values of $m$ we have that $\dim_θP_V(E)=m$ or $\dim_θP_V(E)=\dim_θE$ for $γ_{d,m}-$almost all $V\in G(d,m)$. Our result can be extended to more general functions that include orthogonal projections and fractional Brownian motion. As a particular case, letting $θ=1$, the results are valid for the Box dimension.

math.CA↗

arXiv2024

Intermediate dimensions of complementary sets

Given a positive, non-increasing sequence $a$ with finite sum equal to $1$, we consider the family of all closed subsets of $[0,1]$ whose complementary open intervals have lengths given by a rearrangement of the sequence $a$. We study the full range of possible $\theta$-intermediate dimensions of these sets and, under suitable assumptions on the sequence, we show that this range forms a closed interval, whose endpoints we compute explicitly. This paper fills a gap in the literature concerning the dimensional properties of complementary sets.

math.CA↗

arXiv2024

Weaving Riesz Bases

This paper explores woven frames in separable Hilbert spaces with an initial focus on the finite-dimensional case. We begin by simplifying the problem to bases, for which we obtain a unique characterization. We establish a condition that is both necessary and sufficient for vector reconstruction, which applies to Fourier matrices. Furthermore, we show that these characterizations are still valid in the infinite-dimensional case, for Riesz bases. Finally, we obtain several results for weaving Riesz bases of translations.

math.FA↗

arXiv2024

Periodic Source Detection in Discrete Dynamical Systems via space-time sampling

In this paper, we examine a discrete dynamical system defined by x(n+1) = Ax(n)+ w(n), where x takes values in a Hilbert space H and w is a periodic source with values in a fixed closed subspace W of H. Our goal is to identify conditions on some spatial sampling system G = {gj: j in J} of H that enable stable recovery of the unknown source term w from space-time samples { : n >=0,j in J}. We provide necessary and sufficient conditions on G = {g_j }_{j in J} to ensure stable recovery of any w in W . Additionally, we explicitly construct an operator R, dependent on G, such that R{ }_n,j} = w.

math.CA↗

arXiv2023

Learning optimal smooth invariant subspaces for data approximation

In this article, we consider the problem of approximating a finite set of data (usually huge in applications) by invariant subspaces generated through a small set of smooth functions. The invariance is either by translations under a full-rank lattice or through the action of crystallographic groups. Smoothness is ensured by stipulating that the generators belong to a Paley-Wiener space, that is selected in an optimal way based on the characteristics of the given data. To complete our investigation, we analyze the fundamental role played by the lattice in the process of approximation.

math.OC↗

arXiv2023

Frames of iterations and vector-valued model spaces

Let T be a bounded operator on a Hilbert space H, and F = {f_j: j in J} an at most countable set of vectors in H. In this note, we characterize the pairs {T, F} such that {T^n f: f in F, n in I} form a frame of H, for the cases of I = N_0 and I = Z. The characterization for unilateral iterations gives a similarity with the compression of the shift acting on model spaces of the Hardy space of analytic functions defined on the unit disk with values in $l^2(J). This generalizes recent work for iterations of a single function. In the case of bilateral iterations, the characterization is by the bilateral shift acting on doubly invariant subspaces of L^2(T,l^2(J)). Furthermore, we characterize the frames of iterations for vector-valued model operators when J is finite in terms of Toeplitz and multiplication operators in the unilateral and bilateral case, respectively. Finally, we study the problem of finding the minimal number of orbits that produce a frame in this context.

math.FA↗

arXiv2021

The structure of group preserving operators

In this paper, we prove the existence of a particular diagonalization for normal bounded operators defined on subspaces of $L^2(\mathfrak{S})$ where $\mathfrak{S}$ is a second countable LCA group. The subspaces where the operators act are invariant under the action of a group $Γ$ which is a semi-direct product of a uniform lattice of $\mathfrak{S}$ with a discrete group of automorphisms. This class includes the crystal groups which are important in applications as models for images. The operators are assumed to be $Γ$ preserving. i.e. they commute with the action of $Γ$. In particular we obtain a spectral decomposition for these operators. This generalizes recent results on shift-preserving operators acting on lattice invariant subspaces where $\mathfrak{S}$ is the Euclidean space.

math.FA↗

arXiv2020

Multi-orbital frames through model spaces

We characterize the normal operators $A$ on $\ell^2$ and the elements $a^i \in \ell^2$, with $1\le i\le m$, such that the sequence $$\{ A^n a^1 , \ldots , A^n a^m \}_{n\ge 0}$$ is a frame. The characterization makes strong use of the pseudo-hyperbolic metric of $\mathbb{D}$ and is given in terms of the backward shift invariant subspaces of $H^2(\mathbb{D})$ associated to finite products of interpolating Blaschke products.

math.FA↗

arXiv2020

Continuous and discrete dynamical sampling

In this paper we study the continuous dynamical sampling problem at infinite time in a complex Hilbert space $\mathcal{H}$. We find necessary and sufficient conditions on a bounded linear operator $A\in\mathcal{B}(\mathcal{H})$ and a set of vectors $\mathcal{G}\subset \mathcal{H}$, in order to obtain that $\{e^{tA}g\}_{g\in\mathcal{G}, t\in[0,\infty)}$ is a semi-continuous frame for $\mathcal{H}$. We study if it is possible to discretize the time variable $t$ and still have a frame for $\mathcal{H}$. We also relate the continuous iteration $e^{tA}$ on a set $\mathcal{G}$ to the discrete iteration $(A^\prime)^n$ on $\mathcal{G}^\prime$ for an adequate operator $A^\prime$ and set $\mathcal{G}^\prime\subset \mathcal{H}$.

math.FA↗

arXiv2020

Approximation by group invariant subspaces

In this article we study the structure of $Γ$-invariant spaces of $L^2(\bf R)$. Here $\bf R$ is a second countable LCA group. The invariance is with respect to the action of $Γ$, a non commutative group in the form of a semidirect product of a discrete cocompact subgroup of $\bf R$ and a group of automorphisms. This class includes in particular most of the crystallographic groups. We obtain a complete characterization of $Γ$-invariant subspaces in terms of range functions associated to shift-invariant spaces. We also define a new notion of range function adapted to the $Γ$-invariance and construct Parseval frames of orbits of some elements in the subspace, under the group action. These results are then applied to prove the existence and construction of a $Γ$-invariant subspace that best approximates a set of functional data in $L^2(\bf R)$. This is very relevant in applications since in the euclidean case, $Γ$-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing.

math.FA↗

arXiv2020

Dynamical Sampling: a view from control theory

In this contribution we establish a dictionary between terms in two different areas in order to show that many of the topics studied are common ones - just with a different terminology. We further analyze the relations between the discrete-time and continuous-time versions of the problem, using results from both of these fields. We will also differentiate between a discretization of the continuous-time dynamical system and a discrete-time dynamical system itself.

math.DS↗

arXiv2020

Riesz bases of exponentials and the Bohr topology

We provide a necessary and sufficient condition to ensure that a multi-tile $Ω$ of $R^d$ of positive measure (but not necessarily bounded) admits a structured Riesz basis of exponentials for $ L^{2}(Ω)$. New examples are given and this characterization is generalized to abstract locally compact abelian groups.

math.CA↗

arXiv2019

Local-to-global frames and applications to dynamical sampling problem

In this paper we consider systems of vectors in a Hilbert space $\mathcal{H}$ of the form $\{g_{jk}: j \in J, \, k\in K\}\subset \mathcal{H}$ where $J$ and $K$ are countable sets of indices. We find conditions under which the local reconstruction properties of such a system extend to global stable recovery properties on the whole space. As a particular case, we obtain new local-to-global results for systems of type $\{A^ng\}_{g\in\mathcal{G},0\leq n\leq L }$ arising in the dynamical sampling problem.

math.FA↗

arXiv2019

Optimal translational-rotational invariant dictionaries for images

We provide the construction of a set of square matrices whose translates and rotates provide a Parseval frame that is optimal for approximating a given dataset of images. Our approach is based on abstract harmonic analysis techniques. Optimality is considered with respect to the quadratic error of approximation of the images in the dataset with their projection onto a linear subspace that is invariant under translations and rotations. In addition, we provide an elementary and fully self-contained proof of optimality, and the numerical results from datasets of natural images.

eess.IV↗

arXiv2018

A Fourier Frame for the Middle-Third Cantor Measure

In this paper we show that if $μ$ is any locally and uniformly $α$-dimensional measure supported on a $α$-quasi-regular set $E$, then $L^2(μ)$ admits a frame of exponentials. In particular, for the uniform middle third Cantor measure, $μ_C,$ our result shows that there exists a countable set $Λ$ such that $\{e^{2πi t λ}\}_{λ\in Λ}$ is a frame for $L^2(μ_C)$ (i.e. the measure $μ_C$ admits a generalized spectrum), answering an old outstanding question about the existence of a frame of exponentials for the space $L^2(μ_C)$.

math.CA↗

arXiv2018

Approximation by crystal-refinable function

Let $Γ$ be a crystal group in $\mathbb R^d$. A function $φ:\mathbb R^d\longrightarrow \mathbb C$ is said to be {\em crystal-refinable} (or $Γ-$refinable) if it is a linear combination of finitely many of the rescaled and translated functions $φ(γ^{-1}(ax))$, where the {\em translations} $γ$ are taken on a crystal group $Γ$, and $a$ is an expansive dilation matrix such that $aΓa^{-1}\subsetΓ.$ A $Γ-$refinable function $φ: \mathbb R^d \rightarrow \mathbb C$ satisfies a refinement equation $φ(x)=\sum_{γ\inΓ}d_γφ(γ^{-1}(ax))$ with $d_γ\in \mathbb C$. Let $\mathcal S(φ)$ be the linear span of $\{φ(γ^{-1}(x)): γ\in Γ\}$ and $\mathcal{S}^h=\{f(x/h):f\in\mathcal{S(φ)}\}$. One important property of $\mathcal S(φ)$ is, how well it approximates functions in $L^2(\mathbb R^d)$. This property is very closely related to the {\em crystal-accuracy} of $\mathcal S(φ)$, which is the highest degree $p$ such that all multivariate polynomials $q(x)$ of ${\rm degree}(q)<p$ are exactly reproduced from elements in $\mathcal S(φ)$. In this paper, we determine the accuracy $p$ from the coefficients $d_γ$. Moreover, we obtain from our conditions, a characterization of accuracy for a particular lattice refinable vector function $F$, which simplifies the classical conditions.

math.CA↗