arXiv subjects

Ursula Molter

Publications and source records attributed to Ursula Molter.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2018

A generalized version of the 2-microlocal frontier prescription

The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in $\mathbb {R}^2$, provides a complete and profound classification of pointwise singularity. In \cite{Meyer1998}, \cite{GuiJaffardLevy1998} and \cite{LevySeuret2004} the authors show the following: given a monotone concave downward curve in the plane it is possible to exhibit one function (or distribution) such that its 2-microlocal frontier al $x_0$ is the given curve. In this work we are able to unify the previous results, by obtaining a large class of functions (or distributions), that includes the three examples mentioned above, for which the 2-microlocal frontier is the given curve. The three examples above are in this class. Further, if the curve is a line, we characterize all the functions whose 2-microlocal frontier at $x_0$ is the given line.

math.CA↗

arXiv2018

Small Sets containing any Pattern

Given any dimension function $h$, we construct a perfect set $E \subseteq \mathbb{R}$ of zero $h$-Hausdorff measure, that contains any finite polynomial pattern. This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set $E$ in $\mathbb{R}^N$, of $h$-Hausdorff measure zero, such that for any finite set $\{ f_1,\ldots,f_n\}\subseteq \mathcal{F}$, $E$ satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$. We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_σ$ set without isolated points.

math.CA↗

arXiv2017

Time-frequency shift invariance of Gabor spaces generated by integer lattices

We study extra time-frequency shift invariance properties of Gabor spaces. For a Gabor space generated by an integer lattice, we state and prove several characterizations for its time-frequency shift invariance with respect to a finer integer lattice. The extreme cases of full translation invariance, full modulation invariance, and full time-frequency shift invariance are also considered. The results show a close analogy with the extra translation invariance of shift-invariant spaces.

math.CA↗

arXiv2017

Frames of exponentials and sub-multitiles in LCA groups

In this note we investigate the existence of frames of exponentials for $L^2(Ω)$ in the setting of LCA groups. Our main result shows that sub-multitiling properties of $Ω\subset \widehat{G}$ with respect to a uniform lattice $Γ$ of $\widehat{G}$ guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of $Γ$. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames.

math.CA↗

arXiv2016

Crystallographic Multiwavelets in $L^2(R^d)$

We characterize the scaling function of a crystal Multiresolution Analysis in terms of the vector-scaling function for a Multiresolution Analysis associated to a lattice. We give necessary and sufficient conditions in terms of the symbol matrix in order that an associated crystal wavelet basis exists.

math.CA↗

arXiv2014

Time-frequency shift invariance and the Amalgan Balian Low Theorem

We consider smoothness properties of the generator of a principal Gabor space on the real line which is invariant under some additional translation-modulation pair. We prove that if a Gabor system on a lattice with rational density is a Riesz basis for its closed linear span, and if the closed linear span, a Gabor space, has any additional translation-modulation invariance, then its generator cannot decay well in time and in frequency simultaneously.

math.CA↗

arXiv2013

Countable Contraction Maps in Metric Spaces: Invariant Sets and Measures

We consider a complete metric space $(X,d)$ and a countable number of contractive mappings on $X$, $\mathcal{F}=\{F_i:i\in\mathbb N\}$. We show the existence of a {\em smallest} invariant set (with respect to inclusion) for $\mathcal{F}$. If the maps $F_i$ are of the form $F_i(\x) = r_i \boldmath{x} + b_i$ on $X=\mathbb{R}^d$, we can prove a converse of the classic result on contraction maps. Precisely, we can show that for that case, there exists a {\em unique} bounded invariant set if and only if $r = \sup_i r_i$ is strictly smaller than 1. Further, if $ρ= \{ρ_k\}_{k\in \mathbb N}$ is a probability sequence, we show that if there exists an invariant measure for the system $(\mathcal{F},ρ)$, then it's support must be precisely this smallest invariant set. If in addition there exists any {\em bounded} invariant set, this invariant measure is unique - even though there may be more than one invariant set.

math.CA↗

arXiv2012

Small Furstenberg sets

For $α$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $α$ or $F_α$-set if for each direction $e$ in the unit circle there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is greater or equal than $α$. In this paper we show that if $α> 0$, there exists a set $E\in F_α$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2α}\log^{-θ}(\frac{1}{x})$, $θ>\frac{1+3α}{2}$, which improves on the the previously known bound, that $H^β(E) = 0$ for $β>1/2+3/2α$. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for $\h_γ(x)=\log^{-γ}(\frac{1}{x})$, $γ>0$, we construct a set $E_γ\in F_{\h_γ}$ of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any $E\in F_{\h_γ}$, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_γ$.

math.CA↗

arXiv2012

Non-uniform painless decompositions for anisotropic Besov and Triebel-Lizorkin spaces

In this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the Lebesgue spaces $L^p(\Rdst)$, $1<p<+\infty$. The novelty and difficulty of this construction is that we allow for non-lattice translations. We prove that for an arbitrary expansive matrix $A$ and any set $Λ$ - satisfying a certain spreadness condition but otherwise irregular- there exists a smooth window whose translations along the elements of $Λ$ and dilations by powers of $A$ provide an atomic decomposition for the whole range of the anisotropic Triebel-Lizorkin spaces. The generating window can be either chosen to be bandlimited or to have compact support. To derive these results we start with a known general "painless" construction that has recently appeared in the literature. We show that this construction extends to Besov and Triebel-Lizorkin spaces by providing adequate dual systems.

math.FA↗

arXiv2011

Frames by Multiplication

In this note we study frame-related properties of a sequence of functions multiplied by another function. In particular we study frame and Riesz basis properties. We apply these results to sets of irregular translates of a bandlimited function $h$ in $L^2(\R^d)$. This is achieved by looking at a set of exponentials restricted to a set $E \subset \R^d$ with frequencies in a countable set $Λ$ and multiplying it by the Fourier transform of a fixed function $h \in L^2(E)$. Using density results due to Beurling, we prove the existence and give ways to construct frames by irregular translates.

math.CA↗

arXiv2011

Visible and Invisible Cantor sets

In this article we study for which Cantor sets there exists a gauge-function h, such that the h-Hausdorff-measure is positive and finite. We show that the collection of sets for which this is true is dense in the set of all compact subsets of a Polish space X. More general, any generic Cantor set satisfies that there exists a translation-invariant measure mu for which the set has positive and finite mu-measure. In contrast, we generalize an example of Davies of dimensionless Cantor sets (i.e. a Cantor set for which any translation invariant measure is either zero or non-sigma-finite, that enables us to show that the collection of these sets is also dense in the set of all compact subsets of a Polish space X.

math.CA↗

arXiv2011

A Dimension Reduction Scheme for the Computation of Optimal Unions of Subspaces

Given a set of points \F in a high dimensional space, the problem of finding a union of subspaces \cup_i V_i\subset \R^N that best explains the data \F increases dramatically with the dimension of \R^N. In this article, we study a class of transformations that map the problem into another one in lower dimension. We use the best model in the low dimensional space to approximate the best solution in the original high dimensional space. We then estimate the error produced between this solution and the optimal solution in the high dimensional space.

math.CA↗

arXiv2010

Furstenberg sets for a fractal set of directions

In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $α,β\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{αβ}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $β$ and, for each direction $e$ in $L$, there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is equal or greater than $α$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $\dim(E)\ge\max\left\{α+\fracβ{2} ; 2α+β-1\right\}$ for any $E\in F_{αβ}$. In particular we are able to extend previously known results to the ``endpoint'' $α=0$ case.

math.CA↗

arXiv2009

Improving dimension estimates for Furstenberg-type sets

In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets.

math.CA↗

arXiv2008

Invariance of a Shift-Invariant Space

A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations.

math.FA↗

arXiv2008

Optimal Non-Linear Models for Sparsity and Sampling

Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for the given data, in a sense defined in the paper, and provides an optimal model for sampling in union of subspaces. The results are proved in a general setting and then applied to the case of low dimensional subspaces of R^N and to infinite dimensional shift-invariant spaces in L^2(R^d). We also present an iterative search algorithm for finding the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary design, signal models for signals with finite rate of innovation, and the subspace segmentation problem.

math.CA↗