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> Volume 1, Issue 4
Replies of question in WD 1.2, topic 6.

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Hans Kuehnel. Guest

Posted: Fri Aug 28, 1992 1:50 pm Subject: Replies of question in WD 1.2, topic 6.




Replies of question in WD 1.2, topic 6.
Hello everybody,
here's the summary of replies on my question about
"wavelets with compact support in Fourierspace", Topic #6 in
Wavelet Digest, Wednesday, August 5, 1992, Volume 1, Issue 2
My original question:
> I am curious to know about wavelet bases with compact support in Fourierspace
> (In analogy to the Haar/Daubechies wavelets, which have compact support in
> the space/time domain). Needless to say they should be localized in the
> space/time domain as well. If you have references on this topic please
> mail them to me and I'll summarize the results
> for the list.
Thanks to all who responded:
> From: Wim Sweldens <sweldens@bigcheese.math.sc.edu>
> From: jalvarez@nmsu.edu (Lolina Alvarez)
> From: victor@kirk.wustl.edu (Mladen Victor Wickerhauser)
The punchline was:
Wavelets with compact support in Fourier space are in fact the original ones.
Y. Meyer was the first to design them. As, e.g., described in
(P. Lemarie and Y.Meyer, Ondelettes et bases Hilbertiennes, Rev.
Mat. Iberoamericana 2 (1986), 118) the scaling function looks something
like (LaTeX notation):
[
hatphi(omega)=left{
egin{array}{ll}
1,
&mbox{for $omegaleq {2piover 3}$}\
cosleft({piover 2}
u({3over 2pi}omega1)
ight),
&mbox{for ${2piover 3} < omega < {4piover 3}$}\
0,
&mbox{otherwise}
end{array}
ight.
]
Here, $
u(x)$ is a $C^k$ function with
$
u(x)=
left{
egin{array}{ll}
0, &mbox{$xleq 0$}\
1, &mbox{$x>1$}
end{array}
ight.
$
and $
u(x)+
u(1x)=1$.
The case
$
u(x)=
left{
egin{array}{ll}
0, &mbox{$x<{1over 2}$}\
1, &mbox{$xgeq{1over 2}$}
end{array}
ight.
$
corresponds to the Shannon wavelet.
The replies in detail:

> From sweldens@bigcheese.math.sc.edu
>
> Wavelets with compact support in Fourier space are actually older then
> the ones with compact support in time space.
> The most famous ones are:
> 1. the Meyer wavelet: has faster then inverse polynomial decay in time space
> and is infinitely many times differentiable and orthogonal.
> ref: Yves Meyers book "Ondelettes" published by Hermann in Paris (French).
> 2. the Shannon wavelet:
> in this case the scaling function is the famous Shannon sampling function:
> phi(x) = sin(pi x) / (pi x), and the wavelet is
> psi(x) = [ sin(2 pi x)  sin(pi x) ] / ( pi x ).
> They have compact support in Fourier space and are orthogonal in time
> space, the decay however is very poor.

> From jalvarez@NMSU.Edu
>
> Dear Hans, On page 74 of the first volume of his book on waveletsondelettes
> Meyer considers a wavelet basis generated by a function with compactly
> supported Fourier transform. A similar example is presented by David on his
> Lecture Notes # 1465, page 4 and he calls it Meyer's basis.

> From victor@kirk.wustl.edu
>
> Lemarie and Meyer [1] first constructed bandlimited wavelets, and their
> construction has been simplified in [2]. Recent work by Auscher, Bonami,
> Soria and Weiss in `Revista matematica iberoamericana' has given a complete
> characterization.
>
> [1] Meyer, "Ondelettes et operateurs, Vols I,II", Hermann, Paris (1990)
> [2] Ausher, Weiss, Wickerhauser, "Local sine and cosine bases...", in
> "Waveletsa tutorial in theory and applications", ed. Chui, Academic Press,
> New York (1992)

If someone has some more references on the subject I'd be happy to receive
them.
Regards,
Hans Kuehnel (kuehnel@physik.tumuenchen.de, Tel. +498932093766) 





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