Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups

We prove the existence of sampling sets and interpolation sets near the critical density, in Paley Wiener spaces of a locally compact abelian (LCA) group G . This solves a problem left by Gr\"ochenig, Kutyniok, and Seip in the article: `Landau's density conditions for LCA groups ' (J. of Funct. Anal. 255 (2008) 1831-1850). To achieve this result, we prove the existence of universal Riesz bases of characters for L2(Omega), provided that the relatively compact subset Omega of the dual group of G satisfies a multi-tiling condition. This last result generalizes Fuglede's theorem, and extends to LCA groups setting recent constructions of Riesz bases of exponentials in bounded sets of Rd.Comment: 21 pages, 1 figur

Linear combinations of frame generators in systems of translates

A finitely generated shift invariant space $V$ is a closed subspace of $L^2(\R^d)$ that is generated by the integer translates of a finite number of functions. A set of frame generators for $V$ is a set of functions whose integer translates form a frame for $V$. In this note we give necessary and sufficient conditions in order that a minimal set of frame generators can be obtained by taking linear combinations of the given frame generators. Surprisingly the results are very different to the recently studied case when the property to be a frame is not required.Comment: 13 pages, To appear in J. Math. Anal. App

An Approximation Problem in Multiplicatively Invariant Spaces

Let $\mathcal{H}$ be Hilbert space and $(\Omega,\mu)$ a $\sigma$-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega, \mathcal{H})$ that are invariant under point-wise multiplication by functions in a fix subset of $L^{\infty}(\Omega).$ Given a finite set of data $\mathcal{F}\subseteq L^2(\Omega, \mathcal{H}),$ in this paper we prove the existence and construct an MI space $M$ that best fits $\mathcal{F}$, in the least squares sense. MI spaces are related to shift invariant (SI) spaces via a fiberization map, which allows us to solve an approximation problem for SI spaces in the context of locally compact abelian groups. On the other hand, we introduce the notion of decomposable MI spaces (MI spaces that can be decomposed into an orthogonal sum of MI subspaces) and solve the approximation problem for the class of these spaces. Since SI spaces having extra invariance are in one-to-one relation to decomposable MI spaces, we also solve our approximation problem for this class of SI spaces. Finally we prove that translation invariant spaces are in correspondence with totally decomposable MI spaces.Comment: 18 pages, To appear in Contemporary Mathematic

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.Comment: 15 pages. Some corrections were added, in particular the title was changed. It will appear in "Sampling Theory in Signal and Image Processing