101 research outputs found
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 is a closed subspace of
that is generated by the integer translates of a finite number of
functions. A set of frame generators for is a set of functions whose
integer translates form a frame for . 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 be Hilbert space and a -finite
measure space. Multiplicatively invariant (MI) spaces are closed subspaces of that are invariant under point-wise multiplication by
functions in a fix subset of Given a finite set of data
in this paper we prove the
existence and construct an MI space that best fits , 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
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