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   -> Volume 2, Issue 18

Two papers available
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Author Message
Debao Chen, The University of Texas at Austin,

PostPosted: Fri Nov 29, 2002 3:40 pm    
Subject: Two papers available
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Two papers available

The following two papers are parts of my dissertation. If interested, please
send me e-mail.

Debao Chen
Department of Mathematics
The University of Texas at Austin
Austin, TX 78712

Extended Families of Cardinal Spline wavelets
(This paper has been revised several times. The title has been changed two
times. However, this is the final version and will appear in Applied and
Computational Harmonic Analysis>)

Cardinal spline pre-wavelets $psi_m$ and $eta_m$ are well known.
In this paper we present an extension of cardinal spline pre-wavelets
and construct non-orthogonal cardinal spline
wavelet systems. Both compactly supported spline wavelets
$psi_{m,l;c}$ and globally supported spline
wavelets $eta_{m,l;c} (x) =L_{m+l;c}^{(l)} (2x-1)$
are given. When $l=m $ and $c=0$,
we obtain cardinal spline pre-wavelets $psi_m$ and $eta_m$.
As $l$ decreases, so does the support of the wavelet $psi_{m,l;c}$. When $l$
increases, the smoothness of the dual wavelets $widetilde {psi}_{m,l;c}$
and $widetilde {eta}_{m,l;c}$ improves. When $c=0$ the wavelets
$psi_{m,l;0} = psi_{m,l} $ and $eta_{m,l;0} = eta_{m,l} $
are symmetric or anti-symmetric.
The dual wavelets
$widetilde{psi}_{m,l;0} = widetilde{psi}_{m,l}$ and
$widetilde{eta}_{m,l;0}= widetilde{eta}_{m,l}$ are $l^{ oman{th}}$-order
spline functions. We give an explicit analytic formula of the dual wavelet
The wavelets $psi_{m,l} $ and $eta_{m,l} $ have properties similar to
spline pre-wavelets $psi_m$ and $eta_m$
except for the orthogonality among the wavelet spaces.
Each wavelet is
constructed by spline multiresolution analysis.
The dual MRA's are given.

Spline Wavelets of Small support
(To appear in SIAM J. Anal.)

Every $m^{ oman{th}}$-order spline wavelet
is a linear combination of the functions ${ N_{m+l}^{(l)} (2 x - j), jin
old Z } $.
In this paper we prove that the single function $N_{m+l}^{(l)} (2 x)$,
or $N_{m+l}^{(l)} (2 x - 1)$
is a wavelet when $m$ and $l$ satisfy some mild conditions.
As $l$ decreases, so does the support of the
When $l$ increases,
smoothness of the dual wavelet
Each wavelet is constructed by spline multiresolution analysis.
The dual multiresolution analyses are given.
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