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This
paper
presents a framework for discrete-time signal reconstruction from
absolute values of its short-time Fourier coefficients. Our approach
has two steps. In step one we reconstruct a band-diagonal matrix associated
to
the rank-one operator $K_x=xx^*$. In step two we recover the signal
$x$ by solving an optimization problem. The two steps are somewhat
independent, and one purpose of this talk
is to present a framework that decouples the two problems. The solution
to the first step is connected to the problem of constructing frames
for spaces of Hilbert-Schmidt operators. The second step is somewhat
more elusive. Due to inherent redundancy in recovering $x$ from its
associated rank-one operator $K_x$, the reconstruction problem allows
for imposing supplemental conditions. In this paper we make one such
choice that yields a fast and robust reconstruction. However this
choice may not necessarily be optimal in other situations. It is worth
mentioning that this second step is related to the problem of finding a
rank-one approximation to a matrix with missing data.

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.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.

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.

Shift-invariant
Spaces
in
the Fractional Fourier Transform Domain

In this talk we first introduce the factional Fourier transform and some of its applications in optics, and then proceed to discuss properties of shift-invariant spaces in the fractional Fourier transform domain.

In this talk we first introduce the factional Fourier transform and some of its applications in optics, and then proceed to discuss properties of shift-invariant spaces in the fractional Fourier transform domain.