4,388 research outputs found
Pairs of Frequency-based Nonhomogeneous Dual Wavelet Frames in the Distribution Space
In this paper, we study nonhomogeneous wavelet systems which have close
relations to the fast wavelet transform and homogeneous wavelet systems. We
introduce and characterize a pair of frequency-based nonhomogeneous dual
wavelet frames in the distribution space; the proposed notion enables us to
completely separate the perfect reconstruction property of a wavelet system
from its stability property in function spaces. The results in this paper lead
to a natural explanation for the oblique extension principle, which has been
widely used to construct dual wavelet frames from refinable functions, without
any a priori condition on the generating wavelet functions and refinable
functions. A nonhomogeneous wavelet system, which is not necessarily derived
from refinable functions via a multiresolution analysis, not only has a natural
multiresolution-like structure that is closely linked to the fast wavelet
transform, but also plays a basic role in understanding many aspects of wavelet
theory. To illustrate the flexibility and generality of the approach in this
paper, we further extend our results to nonstationary wavelets with real
dilation factors and to nonstationary wavelet filter banks having the perfect
reconstruction property
Nonhomogeneous Wavelet Systems in High Dimensions
It is of interest to study a wavelet system with a minimum number of
generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in
[11] that for any real-valued expansive matrix M, a homogeneous
orthonormal M-wavelet basis can be generated by a single wavelet function. On
the other hand, it has been demonstrated in [21] that nonhomogeneous wavelet
systems, though much less studied in the literature, play a fundamental role in
wavelet analysis and naturally link many aspects of wavelet analysis together.
In this paper, we are interested in nonhomogeneous wavelet systems in high
dimensions with a minimum number of generators. As we shall see in this paper,
a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system
with almost all properties preserved. We also show that a nonredundant
nonhomogeneous wavelet system is naturally connected to refinable structures
and has a fixed number of wavelet generators. Consequently, it is often
impossible for a nonhomogeneous orthonormal wavelet basis to have a single
wavelet generator. However, for redundant nonhomogeneous wavelet systems, we
show that for any real-valued expansive matrix M, we can always
construct a nonhomogeneous smooth tight M-wavelet frame in with a
single wavelet generator whose Fourier transform is a compactly supported
function. Moreover, such nonhomogeneous tight wavelet frames are
associated with filter banks and can be modified to achieve directionality in
high dimensions. Our analysis of nonhomogeneous wavelet systems employs a
notion of frequency-based nonhomogeneous wavelet systems in the distribution
space. Such a notion allows us to separate the perfect reconstruction property
of a wavelet system from its stability in function spaces
On the lack of compactness on stratified Lie groups
In , the characterization of the \mbox{lack of compactness of
the continuous Sobolev injection }, with
and $\displaystyle{
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
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