4,478 research outputs found
The Real and Complex Techniques in Harmonic Analysis from the Point of View of Covariant Transform
This note reviews complex and real techniques in harmonic analysis. We
describe a common source of both approaches rooted in the covariant transform
generated by the affine group.
Keywords: wavelet, coherent state, covariant transform, reconstruction
formula, the affine group, ax+b-group, square integrable representations,
admissible vectors, Hardy space, fiducial operator, approximation of the
identity, maximal functions, atom, nucleus, atomic decomposition, Cauchy
integral, Poisson integral, Hardy--Littlewood maximal functions, grand maximal
function, vertical maximal functions, non-tangential maximal functions,
intertwining operator, Cauchy-Riemann operator, Laplace operator, singular
integral operator, SIO, boundary behaviour, Carleson measure.Comment: 31 pages, AMS-LaTeX, no figures; v2: a major revision, sections on
representations of the ax+b group and transported norms are added; v3: major
revision: an outline section on complex and real variables techniques are
added, numerous smaller improvements; v4: minor correction
Diffusive wavelets on groups and homogeneous spaces
The aim of this exposition is to explain basic ideas behind the concept of
diffusive wavelets on spheres in the language of representation theory of Lie
groups and within the framework of the group Fourier transform given by
Peter-Weyl decomposition of for a compact Lie group .
After developing a general concept for compact groups and their homogeneous
spaces we give concrete examples for tori -which reflect the situation on
- and for spheres and .Comment: 20 pages, 3 figure
The wavelet transforms in Gelfand-Shilov spaces
We describe local and global properties of wavelet transforms of
ultradifferentiable functions. The results are given in the form of continuity
properties of the wavelet transform on Gelfand-Shilov type spaces and their
duals. In particular, we introduce a new family of highly time-scale localized
spaces on the upper half-space. We study the wavelet synthesis operator (the
left-inverse of the wavelet transform) and obtain the resolution of identity
(Calder\'{o}n reproducing formula) in the context of ultradistributions
ENO-wavelet transforms for piecewise smooth functions
We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate
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