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   -> Volume 3, Issue 11


Question: Mistake in Daubechies' paper ?
 
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steve@pitacat.math.cwru.edu (Steven H. Izen)
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PostPosted: Mon Dec 02, 2002 1:12 pm    
Subject: Question: Mistake in Daubechies' paper ?
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Question: Mistake in Daubechies' paper ?

Q. Is there an error in propostion 2.11 of Daubchies' paper,``The
Wavelet Transform, Time-frequency Localization and Signal Analysis,"
IEEE Trans. Inf. Theory, 36(5), 1990?

I have prepared a one page LaTeX document (below) explaining where I think
the error might be. (I believe it comes down to the Cauchy-Schwarz
inequality being applied in the wrong direction).

Thanks for any help.

Steve

Steve Izen, Associate Prof. of Mathematics, Case Western Reserve University
Best Address: shi@po.cwru.edu Next best address: steve@pitacat.math.cwru.edu
Phones: Analog (Voice): (216)368-2891 <---> Digital (Fax): (216)368-5163

--
documentstyle{article}
egin{document}
In Daubechies' paper ``The Wavelet Transform, Time-frequency
Localization and Signal Analysis", in propositions 2.11 and 2.12 (pages
990-992) conditions are given for frames in Sobolev spaces. The
construction is a generalization of the corresponding construction
for $L^2$. (See Section 3.3.2 of Daubechies' ``Ten Lectures on
Wavelets.")

However, either there is an error in the proof, or I am missing
something. From the last line on page 990,
egin{eqnarray*}
langle f,{T}f angle &=&frac{2pi}{q_0}int
dxleft[(1+x^2)^{s/2}left|hat
f(x) ight| ight]left[(1+x^2)^{-s/2}left|hat
f(x) ight| ight]cdot\
&&cdot
left(sum_mleft|hat g(x+mp_0) ight|^2 ight) + r
end{eqnarray*}



oindent and the bound for $r$ computed on the next page, $|r|leq
frac{2pi}{q_0}R||f||_s||f||_{-s}$, the lower bound
$$
frac{2pi}{q_0}left(inf_xleft(sum_mleft|hat
g(x+mp_0) ight|^2 ight) - R ight)||f||_s||f||_{-s} leq langle
f,{T}f angle
$$
is deduced.

It appears to me that the first term in the left hand side of the
inequality is obtained by writing
egin{eqnarray*}
lefteqn{int
dxleft[(1+x^2)^{s/2}left|hat
f(x) ight| ight]left[(1+x^2)^{-s/2}left|hat f(x) ight| ight]
left(sum_mleft|hat g(x+mp_0) ight|^2 ight)}\
&geq&inf_xleft(sum_mleft|hat g(x+mp_0) ight|^2 ight)int
dxleft[(1+x^2)^{s/2}left|hat
f(x) ight| ight]left[(1+x^2)^{-s/2}left|hat
f(x) ight| ight]. \
end{eqnarray*}
If one had the inequality
$$
int
dxleft[(1+x^2)^{s/2}left|hat
f(x) ight| ight]left[(1+x^2)^{-s/2}left|hat
f(x) ight| ight]geq ||f||_s||f||_{-s},
$$
then the result would follow, but Cauchy-Schwarz gives the inequality
in the opposite direction.

Thus, my question is, have I missed something? Is there anyway to fix
up the argument to obtain a legitimate lower bound? As a side
comment, for $L^2$, the argument goes through with no trouble since
the last step above isn't needed.
end{document}
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