1,015 research outputs found
Shannon Multiresolution Analysis on the Heisenberg Group
We present a notion of frame multiresolution analysis on the Heisenberg
group, abbreviated by FMRA, and study its properties. Using the irreducible
representations of this group, we shall define a sinc-type function which is
our starting point for obtaining the scaling function. Further, we shall give a
concrete example of a wavelet FMRA on the Heisenberg group which is analogous
to the Shannon
MRA on \RR.Comment: 17 page
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
Sampling and Reconstruction of Signals in a Reproducing Kernel Subspace of
In this paper, we consider sampling and reconstruction of signals in a
reproducing kernel subspace of L^p(\Rd), 1\le p\le \infty, associated with an
idempotent integral operator whose kernel has certain off-diagonal decay and
regularity. The space of -integrable non-uniform splines and the
shift-invariant spaces generated by finitely many localized functions are our
model examples of such reproducing kernel subspaces of L^p(\Rd). We show that
a signal in such reproducing kernel subspaces can be reconstructed in a stable
way from its samples taken on a relatively-separated set with sufficiently
small gap. We also study the exponential convergence, consistency, and the
asymptotic pointwise error estimate of the iterative approximation-projection
algorithm and the iterative frame algorithm for reconstructing a signal in
those reproducing kernel spaces from its samples with sufficiently small gap
Slanted matrices, Banach frames, and sampling
In this paper we present a rare combination of abstract results on the
spectral properties of slanted matrices and some of their very specific
applications to frame theory and sampling problems. We show that for a large
class of slanted matrices boundedness below of the corresponding operator in
for some implies boundedness below in for all . We use
the established resultto enrich our understanding of Banach frames and obtain
new results for irregular sampling problems. We also present a version of a
non-commutative Wiener's lemma for slanted matrices
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
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