218 research outputs found
Multiresolution Analysis and Haar Wavelets on the Laguerre Hypergroup
Let ℍn be the Heisenberg group. The fundamental manifold of the radial function space for
ℍn can be denoted by [0,+∞)×ℝ, which is just the Laguerre hypergroup. In this paper the
multiresolution analysis on the Laguerre hypergroup 𝕂=[0,+∞)×ℝ is defined. Moreover the
properties of Haar wavelet bases for La2(𝕂) are investigated
Geometric Aspects of Frame Representations of Abelian Groups
We consider frames arising from the action of a unitary representation of a
discrete countable abelian group. We show that the range of the analysis
operator can be determined by computing which characters appear in the
representation. This allows one to compare the ranges of two such frames, which
is useful for determining similarity and also for multiplexing schemes. Our
results then partially extend to Bessel sequences arising from the action of
the group. We apply the results to sampling on bandlimited functions and to
wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two
sampling transforms to have orthogonal ranges, and two analysis operators for
wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient
condition is easy to compute in terms of the periodization of the Fourier
transform of the frame generators.Comment: 20 pages; contact author: Eric Webe
Shannon Multiresolution Analysis on the Heisenberg Group
We present a notion of frame multiresolution analysis on the Heisenberg
group, abbreviated by FMRA, and study its properties. Using the irreducible
representations of this group, we shall define a sinc-type function which is
our starting point for obtaining the scaling function. Further, we shall give a
concrete example of a wavelet FMRA on the Heisenberg group which is analogous
to the Shannon
MRA on \RR.Comment: 17 page
Random Forests and Networks Analysis
D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient
algorithm based on loop-erased random walks to sample uniform spanning trees
and more generally weighted trees or forests spanning a given graph. This
algorithm provides a powerful tool in analyzing structures on networks and
along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we
focused on applications of spanning rooted forests on finite graphs. The
resulting main conclusions are reviewed in this paper by collecting related
theorems, algorithms, heuristics and numerical experiments. A first
foundational part on determinantal structures and efficient sampling procedures
is followed by four main applications: 1) a random-walk-based notion of
well-distributed points in a graph 2) how to describe metastable dynamics in
finite settings by means of Markov intertwining dualities 3) coarse graining
schemes for networks and associated processes 4) wavelets-like pyramidal
algorithms for graph signals.Comment: Survey pape
Wavelets and their use
This review paper is intended to give a useful guide for those who want to
apply discrete wavelets in their practice. The notion of wavelets and their use
in practical computing and various applications are briefly described, but
rigorous proofs of mathematical statements are omitted, and the reader is just
referred to corresponding literature. The multiresolution analysis and fast
wavelet transform became a standard procedure for dealing with discrete
wavelets. The proper choice of a wavelet and use of nonstandard matrix
multiplication are often crucial for achievement of a goal. Analysis of various
functions with the help of wavelets allows to reveal fractal structures,
singularities etc. Wavelet transform of operator expressions helps solve some
equations. In practical applications one deals often with the discretized
functions, and the problem of stability of wavelet transform and corresponding
numerical algorithms becomes important. After discussing all these topics we
turn to practical applications of the wavelet machinery. They are so numerous
that we have to limit ourselves by some examples only. The authors would be
grateful for any comments which improve this review paper and move us closer to
the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh
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