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   -> Volume 7, Issue 5


Preprint: Preprints from Q. Sun on asymptotic smoothness
 
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"Dr. Sun Qi yu" (matsunqy@leonis.nus.edu.sg)
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PostPosted: Wed May 20, 1998 1:52 pm    
Subject: Preprint: Preprints from Q. Sun on asymptotic smoothness
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#10 Preprint: Preprints from Q. Sun on asymptotic smoothness

Dear editors of Wavelet Digest,

I enclosed three preprints on asympotic estimate of Sobolev exponent of
some refinable functions.

1. Title: Asymptotic Regularity of Daubechies' Scaling Functions
Authors: K.-S. Lau and Qiyu Sun

2. Title: Regularity of Butterworth Refinable Functions
Authors: Ai Hua Fan and Qiyu Sun

3. Title: Sobolev Exponent Estimate and Asymptotic Regularity of M
Band Daubechies' Scaling Functions
Author: Qiyu Sun

In our papers, we introduce a new method to direct estimate Sobolev
exponent and apply it to the asymptotic estimate of Sobolev exponent
of M band Daubechies' scaling functions and Butterworth refinable
functions.

For the Daubechies' scaling functions, the previous difference between
upper and lower estimate of Sobolev exponent $s_p, 1le ple infty$
obtained by Volker (IEEE Trans. Inform. Th., 38(1992), 872-876),
independently by Cohen and Conze (Revista Mat. Iberoamericana,
8(1992), 527-591) is about $C ln N$ for $M=2$, where $N$ is the
papamater in the construction of Daubechies' scaling functions. The
difference obtained by Bi, Dai and Sun (Construction of compactly
supported M band wavelets, Applied Computational Harmonic Analysis, to
appear) is a bounded constant for $Mge 2$.

In our papers, we introduce a term independent of $0<ple infty$
which has explicit expression and show that the difference between the
term above and the Sobolev exponent $s_p, 0<pleinfty$ of Daubechies'
scaling functions is about $C r^N$ for some constants $C$ and $0<r<1$
independent of the parameter $N$. This also affirms the phenomenon
observed by Cohen and Daubechies (Revista Mat. Iberoamericana,
12(1996), 527-591). The phenomenon is observed by Cohen and Daubechies
through numerical computation for $M=2$. In particular, it is their
paper that inspires us to consider the asymptotic estimate of Sobolev
exponent.

About Butterworth refinable functions, Cohen and Daubechies shows in
their paper (Revista Mat. Iberoamericana, 12(1996), 527-591) by
numerical computation that its Sobolev exponent has the similar
phenomenon as the one of Daubechies' scaling functions. Because the
Butterworth refinable functions have infinite support, the phenomenon
is not so clear as the one of Daubechies' scaling functions. In our
paper, we give the upper bound $Nln3/ln2+ln(1+3^{-N})/ln 2$ and $
Nln3/ln2-ln(1+r^N)/ln 2$ of Sobolev exponent $s_p, 0<ple infty$
of Butterworth refinable functions and affirm the phenomenon for
Butterworth refinable functions, where $0<r<1$ and $N$ is the
parameter in the construction of Butterworth refinable functions. By
our computation, $r$ can be chosen as $0.9787028^p$.

The second and third paper can be download from

http://haar.math.nus.edu.sg/~xiatao/sun/SQY1097.ps.Z

http://haar.math.nus.edu.sg/~xiatao/sun/Mregular.ps.Z

With best wishes

Qiyu Sun

Email Address: matsunqy@leonis.nus.edu.sg
Fax: 65-7995452
Homepage: http://haar.math.nus.sg/qsun
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