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   -> Volume 7, Issue 7


Preprint: Preprints concerning the shiftability problem
 
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qiaowang (qiaowang@seu.edu.cn)
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PostPosted: Mon Jul 20, 1998 3:10 am    
Subject: Preprint: Preprints concerning the shiftability problem
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#5 Preprint: Preprints concerning the shiftability problem

We have three preprints concerning the shiftability problems of
wavelet bases and more general configuration. The following are the
abstracts of them. If you need the preprints, please send me an e-mail.

Qiao WANG e-mail: qiaowang@seu.edu.cn

[1] The shiftability of some wavelet bases. by Qiao WANG (submitted)

The wavelet orthonormal bases lack the shiftability. We construct in
this paper a metrical functional for the shiftability of wavelet
bases, where we say $psi$ is shiftable if for every real $s$,
$psi(x-s)in Span { psi(x-n);nin }$, and the functional is $$
r(psi)={1over 2pi}int^{+infty}_{-infty}|widehat psi(xi)|^4
,dxi $$ where the wavelet bases is orthonormal, and the functional
is easy to be generalized to non-orthonormal case by Wigner procedure.
The main result is:

(1) The range of $r(psi)$ is $(0,1]$.
(2) the wavelet base is shiftable iff $r(psi)=1$.
(3) It is nearly shiftable iff it approximates $1$.

Depending on this functional, we researched the shiftability of
B-spline wavelets and Y.Meyer's wavelet, in particular the relation
between the shiftability of scaling function space and that of wavelet
subspace. Some interesting results are proved.

[2] Translation invariance and sampling theorem of wavelet,
by Qiao WANG and Lenan WU (submitted)

The sampling theorem is used at least for two goals: one is to present
a signal, the another is to recover the waveform. The wavelet sampling
theorem systematically researched by Walter and Janssen is applicable
for the first purpose, but not for the second unless the wavelet is
shiftable (Shannon's scaling function) or weak translation invariant
(e.g. Meyer's wavelet). In this correspondence we eneralize our metric
functional, and define the degenerate shiftability. We give a family
of functionals $$r_k(phi)={1over
2pi}int^{+infty}_{-infty}|widehat phi(xi)|^2 |widehat
phi(2^{-k}xi)|^2 ,dxi $$ We proved that: $0<r_0(phi)le
r_1(phi)le r_2(phi)le cdots le 1$ and $lim_{k ends +infty
}r_k(phi)=1$. Thus if $r_n(phi)approx 1$ very closely, one can
recover the waveform by the formula $$ f(x+s)approx sum_k f({kover
2^n}+s) S(x-{kover 2^n})$$ provided $f(x)in V_0=Span{
phi(x-k);kin Z}$ is a finite sum. Using this functional, we found
that $r_1(phi_M)=1$ for Meyer's wavelet which is known by Walter in
another way. And we found that all the B-spline scaling function of
order $n>2$ satisfy $r_1(phi_n)approx 1$ and we construct a group of
shiftable real orthonormal wavelet bases other than Shannon's ones.

[3] A unified approach to shiftability problem,
by Qiao WANG and Lenan WU

For the translation invariance problem concerning translation, dilation
and other unitary operators, we build the metric functional based on the
Fourier analysis on local compact abelian groups. We also give the tile
characterization of the spectrum measure.
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