47,256 research outputs found
Intertwining wavelets or Multiresolution analysis on graphs through random forests
We propose a new method for performing multiscale analysis of functions
defined on the vertices of a finite connected weighted graph. Our approach
relies on a random spanning forest to downsample the set of vertices, and on
approximate solutions of Markov intertwining relation to provide a subgraph
structure and a filter bank leading to a wavelet basis of the set of functions.
Our construction involves two parameters q and q'. The first one controls the
mean number of kept vertices in the downsampling, while the second one is a
tuning parameter between space localization and frequency localization. We
provide an explicit reconstruction formula, bounds on the reconstruction
operator norm and on the error in the intertwining relation, and a Jackson-like
inequality. These bounds lead to recommend a way to choose the parameters q and
q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure
Orthonormal sequences in and time frequency localization
We study uncertainty principles for orthonormal bases and sequences in
. As in the classical Heisenberg inequality we focus on the product
of the dispersions of a function and its Fourier transform. In particular we
prove that there is no orthonormal basis for for which the time and
frequency means as well as the product of dispersions are uniformly bounded.
The problem is related to recent results of J. Benedetto, A. Powell, and Ph.
Jaming.
Our main tool is a time frequency localization inequality for orthonormal
sequences in . It has various other applications.Comment: 18 page
Robust Localization of the Best Error with Finite Elements in the Reaction-Diffusion Norm
We consider the approximation in the reaction-diffusion norm with continuous
finite elements and prove that the best error is equivalent to a sum of the
local best errors on pairs of elements. The equivalence constants do not depend
on the ratio of diffusion to reaction. As application, we derive local error
functionals that ensure robust performance of adaptive tree approximation in
the reaction-diffusion norm.Comment: 21 pages, 1 figur
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