769 research outputs found
Orthogonal Wavelets via Filter Banks: Theory and Applications
Wavelets are used in many applications, including image processing, signal analysis and seismology. The critical problem is the representation of a signal using a small number of computable functions, such that it is represented in a concise and computationally efficient form. It is shown that wavelets are closely related to filter banks (sub band filtering) and that there is a direct analogy between multiresolution analysis in continuous time and a filter bank in discrete time. This provides a clear physical interpretation of the approximation and detail spaces of multiresolution analysis in terms of the frequency bands of a signal. Only orthogonal wavelets, which are derived from orthogonal filter banks, are discussed. Several examples and applications are considered
Classification of Generalized Multiresolution Analyses
We discuss how generalized multiresolution analyses (GMRAs), both classical
and those defined on abstract Hilbert spaces, can be classified by their
multiplicity functions and matrix-valued filter functions . Given a
natural number valued function and a system of functions encoded in a
matrix satisfying certain conditions, a construction procedure is described
that produces an abstract GMRA with multiplicity function and filter
system . An equivalence relation on GMRAs is defined and described in terms
of their associated pairs . This classification system is applied to
classical examples in as well as to previously studied
abstract examples.Comment: 18 pages including bibliograp
Hilbert spaces built on a similarity and on dynamical renormalization
We develop a Hilbert space framework for a number of general multi-scale
problems from dynamics. The aim is to identify a spectral theory for a class of
systems based on iterations of a non-invertible endomorphism.
We are motivated by the more familiar approach to wavelet theory which starts
with the two-to-one endomorphism in the one-torus \bt, a
wavelet filter, and an associated transfer operator. This leads to a scaling
function and a corresponding closed subspace in the Hilbert space
L^2(\br). Using the dyadic scaling on the line \br, one has a nested family
of closed subspaces , n \in \bz, with trivial intersection, and with
dense union in L^2(\br). More generally, we achieve the same outcome, but in
different Hilbert spaces, for a class of non-linear problems. In fact, we see
that the geometry of scales of subspaces in Hilbert space is ubiquitous in the
analysis of multiscale problems, e.g., martingales, complex iteration dynamical
systems, graph-iterated function systems of affine type, and subshifts in
symbolic dynamics. We develop a general framework for these examples which
starts with a fixed endomorphism (i.e., generalizing ) in a
compact metric space . It is assumed that is onto, and
finite-to-one.Comment: v3, minor addition
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
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