Debao Chen, The University of Texas at Austin, Guest

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




Two papers available
The following two papers are parts of my dissertation. If interested, please
send me email.
Debao Chen
Department of Mathematics
The University of Texas at Austin
Austin, TX 78712
email: chen@math.utexas.edu
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>)
Abstract:
Cardinal spline prewavelets $psi_m$ and $eta_m$ are well known.
In this paper we present an extension of cardinal spline prewavelets
and construct nonorthogonal cardinal spline
wavelet systems. Both compactly supported spline wavelets
$psi_{m,l;c}$ and globally supported spline
wavelets $eta_{m,l;c} (x) =L_{m+l;c}^{(l)} (2x1)$
are given. When $l=m $ and $c=0$,
we obtain cardinal spline prewavelets $psi_m$ 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 antisymmetric.
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
$widetilde{psi}_{m,l}$.
The wavelets $psi_{m,l} $ and $eta_{m,l} $ have properties similar to
spline prewavelets $psi_m$ 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.)
Abstract:
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
wavelet.
When $l$ increases,
the
smoothness of the dual wavelet
improves.
Each wavelet is constructed by spline multiresolution analysis.
The dual multiresolution analyses are given. 
