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   -> Volume 7, Issue 8


Preprint: Wavelets and Their Associated Operators (A. Ron)
 
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"Amos Ron" (amos@math.nus.edu.sg)
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PostPosted: Fri Aug 07, 1998 11:23 am    
Subject: Preprint: Wavelets and Their Associated Operators (A. Ron)
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#2 Preprint: Wavelets and Their Associated Operators (A. Ron)

A survey (+ a conjecture) on wavelets and refinable functions

``Wavelets and Their Associated Operators", by Amos Ron, July 1998 (35 pp.)

It is my entry to the Texas IX meeting (i.e, The 9th International
Conference on Appproximation Theory, Nashville, 1/98), and
is actually the first survey that I have ever written.

A .ps file is at

ftp://ftp.cs.wisc.edu/Approx/texas9.ps
or
ftp://ftp.cs.wisc.edu/Approx/texas9.ps.Z

can also be obtained by going to my page
http://www.cs.wisc.edu/~amos
and using the `read me' link.

CONTENT: it is made of two disjoint parts.

the first part overviews the theory of wavelet frames: fiberization of
the shift-invariant system, quasi-wavelet systems, affine products,
the fiberization of the analysis operator of the wavelet system. it
then presents the two extension principles for constructing framelets,
and discusses in detail the tight B-spline framelets. it also shows
how the new theory applies to other topics such as band limited
wavelets and oversampling (as well as Gabor systems)

the second part is devoted to the role of the cascade/transfer
operator in the study of refinable functions. it addresses properties
like the stability of the shifts of a refinable function, the
convergence of the cascade and subdivision algorithms, and the
smoothness of the refinable function. it is written from scratch as a
short monograph, aiming at demonstrating the simplicity and the
elegance of the theory. Highlights in this part are the discrete
cascade operator, the characterization of the smoothness using that
operator, and the accelerations techniques for the cascade iterations
(`the cascade always converges, and fast': one must keep in mind that
Approximation Theory resents results of the form `the cascade operator
converges', unless one establishes the rate of convergence...)

The article ends with a conjecture concerning the convergence of the
cascade iterations. When reading the conjecture, keep in mind that I
have a (simple) proof for the one dimensional case.

Amos Ron, 8/98
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