30 research outputs found
Heisenberg modules as function spaces
Let be a closed, cocompact subgroup of , where
is a second countable, locally compact abelian group. Using localization of
Hilbert -modules, we show that the Heisenberg module
over the twisted group -algebra
due to Rieffel can be continuously and densely embedded into the Hilbert space
. This allows us to characterize a finite set of generators for
as exactly the generators of multi-window
(continuous) Gabor frames over , a result which was previously known
only for a dense subspace of . We show that
as a function space satisfies two properties that
make it eligible for time-frequency analysis: Its elements satisfy the
fundamental identity of Gabor analysis if is a lattice, and their
associated frame operators corresponding to are bounded.Comment: 24 pages; several changes have been made to the presentation, while
the content remains essentially unchanged; to appear in Journal of Fourier
Analysis and Application
From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces
This article describes how the ideas promoted by the fundamental papers
published by M. Frazier and B. Jawerth in the eighties have influenced
subsequent developments related to the theory of atomic decompositions and
Banach frames for function spaces such as the modulation spaces and
Besov-Triebel-Lizorkin spaces.
Both of these classes of spaces arise as special cases of two different,
general constructions of function spaces: coorbit spaces and decomposition
spaces. Coorbit spaces are defined by imposing certain decay conditions on the
so-called voice transform of the function/distribution under consideration. As
a concrete example, one might think of the wavelet transform, leading to the
theory of Besov-Triebel-Lizorkin spaces.
Decomposition spaces, on the other hand, are defined using certain
decompositions in the Fourier domain. For Besov-Triebel-Lizorkin spaces, one
uses a dyadic decomposition, while a uniform decomposition yields modulation
spaces.
Only recently, the second author has established a fruitful connection
between modern variants of wavelet theory with respect to general dilation
groups (which can be treated in the context of coorbit theory) and a particular
family of decomposition spaces. In this way, optimal inclusion results and
invariance properties for a variety of smoothness spaces can be established. We
will present an outline of these connections and comment on the basic results
arising in this context
Gabor Duality Theory for Morita Equivalent -algebras
The duality principle for Gabor frames is one of the pillars of Gabor
analysis. We establish a far-reaching generalization to Morita equivalent
-algebras where the equivalence bimodule is a finitely generated
projective Hilbert -module. These Hilbert -modules are equipped with
some extra structure and are called Gabor bimodules. We formulate a duality
principle for standard module frames for Gabor bimodules which reduces to the
well-known Gabor duality principle for twisted group -algebras of a
lattice in phase space. We lift all these results to the matrix algebra level
and in the description of the module frames associated to a matrix Gabor
bimodule we introduce -matrix frames, which generalize superframes and
multi-window frames. Density theorems for -matrix frames are
established, which extend the ones for multi-window and super Gabor frames. Our
approach is based on the localization of a Hilbert -module with respect to
a trace.Comment: 36 page
Multigenerator Gabor Frames on Local Fields
The main objective of this paper is to provide complete characterization of multigenerator Gabor frames on a periodic set in . In particular, we provide some necessary and sufficient conditions for the multigenerator Gabor system to be a frame for . Furthermore, we establish the complete characterizations of multigenerator Parseval Gabor frames
Quantum Frames and Uncertainty Principles arising from Symplectomorphisms
In this thesis, a new generalized signal transform along with a new uncertainty principle is elaborated. Starting from a coordinate system, associated to a specific symplectomorphism on phase space, the coordinates are used to define curvilinear flows along which the phase space picture of a prototype function is translated. As the uncertainty principle restricts the amount to which the phase space picture of functions can be concentrated, with each such function is associated a phase space cell. Using the phase space translates of these cells, the notion of a quantum frame is defined, by means of which a reservoir of interesting functions may be decomposed. To define optimal phase space cells, two complementing uncertainty principles, associated with coordinate systems in phase space, are introduced, one of which measures the deviation from the chosen frame, while the other optimizes with respect to the canonically conjugate coordinates and leads to more concentrated waveforms
Bifurcation analysis of the Topp model
In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao