Chris
Heil - Georgia Institute of Technology, Atlanta
A
Selection
of
Problems
in
Time-Frequency Analysis and Wavelet Theory
This
talk
will survey a range of open problems in time-frequency analysis
and
wavelet
theory,
ranging
from
esoteric to practical.
Some of the problems include the HRT conjecture on linear independence
of time-frequency shifts, the Olson-Zalik conjecture on Schauder bases
of
translates,
and
frame
bounds
of finite wavelet and Gabor systems.
Emily
King - National Institutes of Health, Washington DC
Representation
theory
of generalized shearlets
A
number
of new transformations have sprung up in an attempt to
detect
directional trends in $2$-$d$ data, including shearlets, for
which
nice
algorithms and theory exist. There is a need to develop higher
dimensional
tools for various applications, like biomedical imaging.
We
exploit the representation theory of the extended metaplectic
group
in order to construct isotropic and anisotropic analogs of
the
shearlet transform over $L^2(R^d)$ for $d \geq 2$. The new representations
are
then analyzed and co-orbit space theory is applied in order
to
create discrete versions of these systems.
Maria
Skopina - St. Petersburg State University, Russia
Multivariate wavelet frames and frame-like
systems
In
order
to construct a wavelet frame with a desirable approximation
order,
it is necessary to provide vanishing moments for the generating wavelet
functions.In
the
multi-dimensional
case
this problem is much more complicated than
its one-dimensional version. In particular, two open algebraic problems
are obstacles for the construction of compactly supported multivariate
tight wavelet frames. Namely, first, it is not known if any appropriate
row
can be extended to a unitary matrix whose entries are trigonometric
polynomials.
Second, it is not known if any non-negative trigonometric polynomial
can be represented as a finite sum of squared magnitude of
trigonometric polynomials.
We suggest a way to get around these obstacles and give a constructive
method
for the improvement an arbitrary appropriate mask to obtain a scaling
mask
generating
a
compactly
supported
tight wavelet frame with a required approximation
order. A
method
for
the
construction of dual wavelet frames is also developed.
It
appears
that
frame-type
decompositions hold for some MRA-based wavelet
systems which
are
not
frames
in L_2. Due to resent results by B.Han and Z.Shen,
it
may
happen
that
such a system is a frame in the Sobolev space. We
study
frame-type
decompositions
and their approximation order in a
more general situation.
