Akram Aldroubi (aldroubi@helix.nih.gov) Guest

Posted: Wed Dec 04, 2002 9:49 am Subject: Preprint: Oblique and biorthogonal multiwavelet bases




Preprint: Oblique and biorthogonal multiwavelet bases
Title: Oblique and biorthogonal multiwavelet bases
Author: Akram Aldroubi
Abstract: We develop the theory of oblique multiwavelet bases, which
encompasses the orthogonal, semiorthogonal and biorthogonal cases.
Oblique multiwavelets preserve the advantages of orthogonal and
biorthogonal wavelets and enhance the flexibility of the theory to
accommodate a wider variety of wavelet bases. We illustrate the
results using the Haar scaling function and the
CohenDaubechiesPlonka multiscaling function.
To obtain a preprint, send an email to aldroubi@helix.nih.gov
and include your postal address.
Title: Oblique projections in atomic spaces
Author: Akram Aldroubi
Journal: Proc. Amer. math. Soc., to appear.
Abstract: Let $HH$ be a Hilbert space, $O$ a unitary operator on
$HH$, and reak ${phi^i}_{i=1,dots,r.}$ $r$ vectors in $HH$.
We construct an {it atomic subspace} $U subset HH$: [ U=left{ {
sumlimits_{i=1,dots,r} {sumlimits_{kin Z}
{c^i(k)O^kphi ^i}:;c^iin l^2,forall i=1,dots,r}}
ight}
] We give the necessary and sufficient conditions for $U$ to be a
welldefined, closed subspace of $HH$ with $left{ {O^kphi ^i}
ight}_{i=1,dots,r. ;kin Z}$ as its Riesz basis. We then
consider the oblique projection $P_{{scriptscriptstyle Uot V}}$
on the space $U(O,{phi^i_{scriptscriptstyle U}}_{i=1,dots,r})$
in a direction orthogonal to $V(O,{phi^i_{scriptscriptstyle
V}}_{i=1,dots,r})$. We give the necessary and sufficient conditions
on $O,{phi^i_{scriptscriptstyle U}}_{i=1,dots,r}$, and
${phi^i_{scriptscriptstyle V}}_{i=1,dots,r}$ for
$P_{{scriptscriptstyle Uot V}}$ to be welldefined. The results
can be used to construct biorthogonal multiwavelets in various
spaces. They can also be used to generalize the ShannonWhittaker
theory on uniform sampling.
To obtain a preprint, send an email to aldroubi@helix.nih.gov and
include a postal address. 
