5,639 research outputs found
Irregular and multi--channel sampling of operators
The classical sampling theorem for bandlimited functions has recently been
generalized to apply to so-called bandlimited operators, that is, to operators
with band-limited Kohn-Nirenberg symbols. Here, we discuss operator sampling
versions of two of the most central extensions to the classical sampling
theorem. In irregular operator sampling, the sampling set is not periodic with
uniform distance. In multi-channel operator sampling, we obtain complete
information on an operator by multiple operator sampling outputs
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
Construction of frames for shift-invariant spaces
We construct a sequence {\phi_i(\cdot-j)\mid j\in{\ZZ}, i=1,...,r} which
constitutes a -frame for the weighted shift-invariant space
[V^p_{\mu}(\Phi)=\Big{\sum\limits_{i=1}^r\sum\limits_{j\in{\mathbb{Z}}}c_i(j)\phi_i(\cdot-j)
\Big| {c_i(j)}_{j\in{\mathbb{Z}}}\in\ell^p_{\mu}, i=1,...,r\Big},
p\in[1,\infty],] and generates a closed shift-invariant subspace of
. The first construction is obtained by choosing functions
, , with compactly supported Fourier transforms
, . The second construction, with compactly supported
gives the Riesz basis
A Hilbert Space Theory of Generalized Graph Signal Processing
Graph signal processing (GSP) has become an important tool in many areas such
as image processing, networking learning and analysis of social network data.
In this paper, we propose a broader framework that not only encompasses
traditional GSP as a special case, but also includes a hybrid framework of
graph and classical signal processing over a continuous domain. Our framework
relies extensively on concepts and tools from functional analysis to generalize
traditional GSP to graph signals in a separable Hilbert space with infinite
dimensions. We develop a concept analogous to Fourier transform for generalized
GSP and the theory of filtering and sampling such signals
Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolution Systems
Let be a finite abelian group and be a circular
convolution operator on . The problem under consideration is how to
construct minimal and such that is
a frame for , where is the canonical
basis of . This problem is motivated by the spatiotemporal sampling
problem in discrete spatially invariant evolution systems. We will show that
the cardinality of should be at least equal to the largest geometric
multiplicity of eigenvalues of , and we consider the universal
spatiotemporal sampling sets for convolution operators
with eigenvalues subject to the same largest geometric
multiplicity. We will give an algebraic characterization for such sampling sets
and show how this problem is linked with sparse signal processing theory and
polynomial interpolation theory
Surgery of spline-type and molecular frames
We prove a result about producing new frames for general spline-type spaces
by piecing together portions of known frames. Using spline-type spaces as
models for the range of certain integral transforms, we obtain results for
time-frequency decompositions and sampling.Comment: 34 pages. Corrected typo
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