31 research outputs found
Continuous Frames, Function Spaces, and the Discretization Problem
A continuous frame is a family of vectors in a Hilbert space which allows
reproductions of arbitrary elements by continuous superpositions. Associated to
a given continuous frame we construct certain Banach spaces. Many classical
function spaces can be identified as such spaces. We provide a general method
to derive Banach frames and atomic decompositions for these Banach spaces by
sampling the continuous frame. This is done by generalizing the coorbit space
theory developed by Feichtinger and Groechenig. As an important tool the
concept of localization of frames is extended to continuous frames. As a
byproduct we give a partial answer to the question raised by Ali, Antoine and
Gazeau whether any continuous frame admits a corresponding discrete realization
generated by sampling.Comment: 44 page
Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type
Coorbit space theory is an abstract approach to function spaces and their
atomic decompositions. The original theory developed by Feichtinger and
Gr{\"o}chenig in the late 1980ies heavily uses integrable representations of
locally compact groups. Their theory covers, in particular, homogeneous
Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the
recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces
cannot be covered by their group theoretical approach. Later it was recognized
by Fornasier and the first named author that one may replace coherent states
related to the group representation by more general abstract continuous frames.
In the first part of the present paper we significantly extend this abstract
generalized coorbit space theory to treat a wider variety of coorbit spaces. A
unified approach towards atomic decompositions and Banach frames with new
results for general coorbit spaces is presented. In the second part we apply
the abstract setting to a specific framework and study coorbits of what we call
Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel
spaces of various types of interest as coorbits. We obtain several old and new
wavelet characterizations based on precise smoothness, decay, and vanishing
moment assumptions of the respective wavelet. As main examples we obtain
results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal
spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed
smoothness, and even mixtures of the mentioned ones. Due to the generality of
our approach, there are many more examples of interest where the abstract
coorbit space theory is applicable.Comment: to appear in Journal of Functional Analysi
Coorbits for projective representations with an application to Bergman spaces
Recently representation theory has been used to provide atomic decompositions
for a large collection of classical Banach spaces. In this paper we extend the
techniques to also include projective representations. As our main application
we obtain atomic decompositions of Bergman spaces on the unit ball through the
holomorphic discrete series for the group of isometries of the ball
Banach frames in coorbit spaces consisting of elements which are invariant under symmetry groups
AbstractThis paper is concerned with the construction of atomic decompositions and Banach frames for subspaces of certain Banach spaces consisting of elements which are invariant under some symmetry group. These Banach spaces—called coorbit spaces—are related to an integrable group representation. The construction is established via a generalization of the well-established Feichtinger–Gröchenig theory. Examples include radial wavelet-like atomic decompositions and frames for radial Besov–Triebel–Lizorkin spaces, as well as radial Gabor frames and atomic decompositions for radial modulation spaces
Quantum tomography with wavelet transform in Banach space on Homogeneous space
The intimate connection between the Banach space wavelet reconstruction
method on homogeneous spaces with both singular and nonsingular vacuum vectors,
and some of well known quantum tomographies, such as: Moyal-representation for
a spin, discrete phase space tomography, tomography of a free particle,
Homodyne tomography, phase space tomography and SU(1,1) tomography is
explained. Also both the atomic decomposition and banach frame nature of these
quantum tomographic examples is explained in details. Finally the connection
between the wavelet formalism on Banach space and Q-function is discussed.Comment: 25 page