28 research outputs found
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
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
Embeddings of Decomposition Spaces into Sobolev and BV Spaces
In the present paper, we investigate whether an embedding of a decomposition
space into a given Sobolev space
exists. As special cases, this includes embeddings
into Sobolev spaces of (homogeneous and inhomogeneous) Besov spaces,
()-modulation spaces, shearlet smoothness spaces and also of a large
class of wavelet coorbit spaces, in particular of shearlet-type coorbit spaces.
Precisely, we will show that under extremely mild assumptions on the covering
, we have
as soon as and
hold. Here, and the weight
can be easily computed, only based on the covering
and on the parameters .
Conversely, a necessary condition for existence of the embedding is that
and
hold, where denotes the space of finitely supported
sequences on .
All in all, for the range , we obtain a complete
characterization of existence of the embedding in terms of readily verifiable
criteria. We can also completely characterize existence of an embedding of a
decomposition space into a BV space
On Homogeneous Decomposition Spaces and Associated Decompositions of Distribution Spaces
A new construction of decomposition smoothness spaces of homogeneous type is
considered. The smoothness spaces are based on structured and flexible
decompositions of the frequency space . We
construct simple adapted tight frames for that can be used
to fully characterise the smoothness norm in terms of a sparseness condition
imposed on the frame coefficients. Moreover, it is proved that the frames
provide a universal decomposition of tempered distributions with convergence in
the tempered distributions modulo polynomials. As an application of the general
theory, the notion of homogeneous -modulation spaces is introduced.Comment: 27 page