39 research outputs found
Linear combinations of generators in multiplicatively invariant spaces
Multiplicatively invariant (MI) spaces are closed subspaces of
that are invariant under multiplications of (some)
functions in . In this paper we work with MI spaces that
are finitely generated. We prove that almost every linear combination of the
generators of a finitely generated MI space produces a new set on generators
for the same space and we give necessary and sufficient conditions on the
linear combinations to preserve frame properties. We then apply what we prove
for MI spaces to system of translates in the context of locally compact abelian
groups and we obtain results that extend those previously proven for systems of
integer translates in .Comment: 13 pages. Minor changes have been made. To appear in Studia
Mathematic
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
Shift-modulation invariant spaces on LCA groups
A shift-modulation invariant space is a subspace of ,
that is invariant by translations along elements in and modulations by
elements in . Here is a locally compact abelian group, and and
are closed subgroups of and the dual group ,
respectively. In this article we provide a characterization of shift-modulation
invariant spaces in this general context when and are uniform
lattices. This extends previous results known for . We develop
fiberization techniques and suitable range functions adapted to LCA groups
needed to provide the desired characterization.Comment: 17 page
Shift Invariant Spaces on LCA Groups
In this article we extend the theory of shift-invariant spaces to the context
of LCA groups. We introduce the notion of H-invariant space for a countable
discrete subgroup H of an LCA group G, and show that the concept of range
function and the techniques of fiberization are valid in this context. As a
consequence of this generalization we prove characterizations of frames and
Riesz bases of these spaces extending previous results, that were known for Rd
and the lattice Zd .Comment: 23 pages. Some small corrections were added. To appear in "Journal of
Functional Analysis