23,695 research outputs found
Uncertainty Relations for Shift-Invariant Analog Signals
The past several years have witnessed a surge of research investigating
various aspects of sparse representations and compressed sensing. Most of this
work has focused on the finite-dimensional setting in which the goal is to
decompose a finite-length vector into a given finite dictionary. Underlying
many of these results is the conceptual notion of an uncertainty principle: a
signal cannot be sparsely represented in two different bases. Here, we extend
these ideas and results to the analog, infinite-dimensional setting by
considering signals that lie in a finitely-generated shift-invariant (SI)
space. This class of signals is rich enough to include many interesting special
cases such as multiband signals and splines. By adapting the notion of
coherence defined for finite dictionaries to infinite SI representations, we
develop an uncertainty principle similar in spirit to its finite counterpart.
We demonstrate tightness of our bound by considering a bandlimited lowpass
train that achieves the uncertainty principle. Building upon these results and
similar work in the finite setting, we show how to find a sparse decomposition
in an overcomplete dictionary by solving a convex optimization problem. The
distinguishing feature of our approach is the fact that even though the problem
is defined over an infinite domain with infinitely many variables and
constraints, under certain conditions on the dictionary spectrum our algorithm
can find the sparsest representation by solving a finite-dimensional problem.Comment: Accepted to IEEE Trans. on Inform. Theor
Orthogonal Ramanujan Sums, its properties and Applications in Multiresolution Analysis
Signal processing community has recently shown interest in Ramanujan sums
which was defined by S.Ramanujan in 1918. In this paper we have proposed
Orthog- onal Ramanujan Sums (ORS) based on Ramanujan sums. In this paper we
present two novel application of ORS. Firstly a new representation of a finite
length signal is given using ORS which is defined as Orthogonal Ramanujan
Periodic Transform.Secondly ORS has been applied to multiresolution analysis
and it is shown that Haar transform is a spe- cial case
A note related to the CS decomposition and the BK inequality for discrete determinantal processes
We prove that for a discrete determinantal process the BK inequality occurs
for increasing events generated by simple points. We give also some elementary,
but nonetheless appealing relationship, between a discrete determinantal
process and the well-known CS decomposition.Comment: To appear in Journal of Applied Probabilit
Closable Hankel operators and moment problems
In a paper from 2016 D. R. Yafaev considers Hankel operators associated with
Hamburger moment sequences q_n and claims that the corresponding Hankel form is
closable if and only if the moment sequence tends to 0. The claim is not
correct, since we prove closability for any indeterminate moment sequence but
also for certain determinate moment sequences corresponding to measures with
finite index of determinacy. It is also established that Yafaev's result holds
if the moments satisfy \root{2n}\of{q_{2n}}=o(n).Comment: 10 pages. The notation for the closure of an operator A is changed to
\overline{A} from \o A on pages 7,
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