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> Volume 7, Issue 1
Question: Wavelets and functional analysis

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Dave Dixon (dixon@agouti.ucr.edu) Guest

Posted: Thu Jan 08, 1998 3:46 am Subject: Question: Wavelets and functional analysis




#23 Question: Wavelets and functional analysis
Hello. I am trying to understand some of the deeper mathematical
aspects of wavelets, particular how they relate to function classes,
etc. However, as a physicist I'm having a hard time getting my head
around the literature, such as Meyer's book. Are there any references
addressing this topic on a conceptual level, maybe assuming no more
than a knowledge of calculus, basic group theory, Fourier analysis,
etc? Something with some basic examples and pictures?
Some specific topics I'd like to get a better handle on:
 What are the various function classes, and where might specific ones prove
interesting (even approximately) in the real world?
 How/why do wavelets form "nice" bases for these function classes?
 How are wavelets related to measures of "smoothness"? How might
these measures be useful in quantifying aspects of a real signal or
image?
 What are the properties of an "unconditional basis"? Why does
unconditionality make us happy?
 What are things like the Wiener algebra, bump algebra, etc., and how
are they related to the above issues?
I realize that's a lot of ground to cover, and much of it may just be
a matter of looking at some understandable references on background
material before tackling Meyer and similar works. Or, if the experts
want to take a crack at answering some of these things in a basic
manner, I'd be willing to collate whatever info I get and put together
a WWW document for the nonexperts. It seems to me that the work of
Donoho and others indicates that knowledge of these aspects is
important.
Thanks!
Dave 





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