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> Volume 7, Issue 9
Thesis: Some new aspects of wavelet and Gabor transform...

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Elke Wilczok (elke@applmath.tumuenchen.de) Guest

Posted: Tue Aug 25, 1998 10:38 am Subject: Thesis: Some new aspects of wavelet and Gabor transform...




#5 Thesis: Some new aspects of wavelet and Gabor transform...
Some months ago I finished my PhD thesis concerning new aspects of
wavelet and Gabor transform as seen from the viewpoint of functional
analysis.
It mainly consists of four parts:
1.) Characterization of wavelet and Gabor transform
by covariance properties
2.) Description of the ranges
3.) Timefrequencylocalization
4.) Uncertainty principles
Unfortunately, the thesis is available in GERMAN only.
It can be downloaded as a psfile from my home page
http://wwwm6.mathematik.tumuenchen.de/~elke/
its German title is
"Zur Funktionalanalysis der Wavelet und Gabortransformation".
Concerning chapter 4 (uncertainty principles) an ENGLISH
preprint can be found on the same website. Its title is
"New Uncertainty Principles for the Continuous Gabor Transform
and the Continuous Wavelet Transform."
An ABSTRACT of this paper is given in the following:
Gabor and wavelet methods are preferred to classical Fourier methods,
whenever the time dependence of the analyzed signal is of the same
importance as its frequency dependence. However, there exist strict
limits to the maximal timefrequency resolution of these both
transforms, similar to Heisenberg's uncertainty principle in Fourier
analysis. Results of this type are the subject of the following
article. Among else, the following will be shown: if psi is a window
function,
f in L^2(R){0} an arbitrary signal and
G_psi f(omega,t) the continuous Gabor transform
of f with respect to psi,
then the support of G_psi f(omega,t), considered as a subset of the
timefrequencyplane R^2, cannot possess finite Lebesgue measure. The
proof of this statement, as well as the proof of its wavelet
counterpart, relies heavily on the well known fact that the ranges of
the continuous transforms are reproducing kernel Hilbert spaces,
showing some kind of shiftinvariance. The last point prohibits the
extension of results of this type to discrete theory.
Elke Wilczok Phone: ++49 89 289 226 25
Zentrum Mathematik Fax: ++49 89 289 226 05
TU Muenchen
Arcisstr.21 email:
80290 Muenchen elke@applmath.tumuenchen.de
GERMANY 





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