16,515 research outputs found
Modulated Unit-Norm Tight Frames for Compressed Sensing
In this paper, we propose a compressed sensing (CS) framework that consists
of three parts: a unit-norm tight frame (UTF), a random diagonal matrix and a
column-wise orthonormal matrix. We prove that this structure satisfies the
restricted isometry property (RIP) with high probability if the number of
measurements for -sparse signals of length
and if the column-wise orthonormal matrix is bounded. Some existing structured
sensing models can be studied under this framework, which then gives tighter
bounds on the required number of measurements to satisfy the RIP. More
importantly, we propose several structured sensing models by appealing to this
unified framework, such as a general sensing model with arbitrary/determinisic
subsamplers, a fast and efficient block compressed sensing scheme, and
structured sensing matrices with deterministic phase modulations, all of which
can lead to improvements on practical applications. In particular, one of the
constructions is applied to simplify the transceiver design of CS-based channel
estimation for orthogonal frequency division multiplexing (OFDM) systems.Comment: submitted to IEEE Transactions on Signal Processin
Fourier-Stieltjes algebras of locally compact groupoids
This paper gives a first step toward extending the theory of
Fourier-Stieltjes algebras from groups to groupoids. If G is a locally compact
(second countable) groupoid, we show that B(G), the linear span of the Borel
positive definite functions on G, is a Banach algebra when represented as an
algebra of completely bounded maps on a C^*-algebra associated with G. This
necessarily involves identifying equivalent elements of B(G). An example shows
that the linear span of the continuous positive definite functions need not be
complete. For groups, B(G) is isometric to the Banach space dual of C^*(G). For
groupoids, the best analog of that fact is to be found in a representation of
B(G) as a Banach space of completely bounded maps from a C^*-algebra associated
with G to a C^*-algebra associated with the equivalence relation induced by G.
This paper adds weight to the clues in the earlier study of Fourier-Stieltjes
algebras that there is a much more general kind of duality for Banach algebras
waiting to be explored.Comment: 34 page
Best possible rates of distribution of dense lattice orbits in homogeneous spaces
The present paper establishes upper and lower bounds on the speed of
approximation in a wide range of natural Diophantine approximation problems.
The upper and lower bounds coincide in many cases, giving rise to optimal
results in Diophantine approximation which were inaccessible previously. Our
approach proceeds by establishing, more generally, upper and lower bounds for
the rate of distribution of dense orbits of a lattice subgroup in a
connected Lie (or algebraic) group , acting on suitable homogeneous spaces
. The upper bound is derived using a quantitative duality principle for
homogeneous spaces, reducing it to a rate of convergence in the mean ergodic
theorem for a family of averaging operators supported on and acting on
. In particular, the quality of the upper bound on the rate of
distribution we obtain is determined explicitly by the spectrum of in the
automorphic representation on . We show that the rate
is best possible when the representation in question is tempered, and show that
the latter condition holds in a wide range of examples
Optimal covariant measurements: the case of a compact symmetry group and phase observables
We study various optimality criteria for quantum observables. Observables are
represented as covariant positive operator valued measures and we consider the
case when the symmetry group is compact. Phase observables are examined as an
example
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