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Series: Analysis Seminar

We introduce a class of operators on abstract measurable spaces, which unifies variety of operators in Harmonic Analysis. We prove that such operators can be dominated by simple sparse operators. Those domination theorems imply some new estimations for Calderón-Zygmund operators, martingale transforms and Carleson operators.

Series: Analysis Seminar

The Gabor system of a function is the set
of all of its integer translations and modulations. The Balian-Low
Theorem states that the Gabor system of a function which is well
localized in both time and frequency cannot form an Riesz basis for
$L^2(\mathbb{R})$.
An important tool in the proof is a characterization of the Riesz basis
property in terms of the boundedness of the Zak transform of the
function. In this talk, we will discuss results showing that weaker
basis-type properties also correspond to boundedness
of the Zak transform, but in the sense of Fourier multipliers. We will
also discuss using these results to prove generalizations of the
Balian-Low theorem for Gabor systems with weaker basis properties, as
well as for shift-invariant spaces with multiple
generators and in higher dimensions.

Series: Analysis Seminar

It is a conjecture of Zygmund that the averages of a square integrable function over line segments oriented along a Lipschitz vector field on the plane converge pointwise almost everywhere. This statement is equivalent to the weak L^2 boundedness of the directional maximal operator along the vector field. A related conjecture, attributed to Stein, is the weak L^2 boundedness of the directional Hilbert transform taken along a Lipschitz vector field. In this talk, we will discuss recent partial progress towards Stein’s conjecture obtained in collaboration with I. Parissis, and separately with S. Guo, C. Thiele and P. Zorin-Kranich. In particular, I will discuss the recently obtained sharp bound for the Hilbert transform along finite order lacunary sets in two dimensions and possible higher dimensional generalization

Series: Analysis Seminar

Magyar, Stein, and Wainger proved a discrete variant in
Zd
of the continuous spherical maximal theorem in
Rd
for all
d ≥
5. Their argument
proceeded via the celebrated “circle method” of Hardy, Littlewood, and
Ramanujan and relied on estimates for continuous spherical maximal
averages via a general transference principle.
In this talk, we introduce a range of sparse bounds for discrete
spherical maximal averages and discuss some ideas needed to obtain satisfactory control on the major
and minor arcs. No sparse bounds were previously known in this setting.

Series: Analysis Seminar

A sparse bound is a novel method to bound a bilinear
form. Such a bound gives effortless weighted inequalities, which are
also easy to quantify. The range of forms which admit a sparse bound is
broad. This short survey of the subject will include the case of
spherical averages, which has a remarkably easy proof.

Series: Analysis Seminar

The
classical Balian-Low theorem states that if both a function and it's
Fourier transform decay too fast then the Gabor system generated by this
function (i.e. the system obtained from this function by taking integer
translations and integer modulations) cannot be an orthonormal basis or a Riesz basis.Though it provides for an
excellent `thumbs--rule' in time-frequency analysis, the Balian--Low
theorem is not adaptable to many applications. This is due to the fact
that in realistic situations information about a signal is given by a
finite dimensional vector rather then by a function over the real line.
In this work we obtain an analog of the Balian--Low theorem in the
finite dimensional setting, as well as analogs to some of its
extensions. Moreover, we will note that the classical Balian--Low
theorem can be derived from these finite dimensional analogs.

Series: Analysis Seminar

Abstract: Shift-invariant (SI) spaces play a prominent role in the study
of wavelets, Gabor systems, and other group frames. Working in the
setting of LCA groups, we use a variant of the Zak transform to classify
SI spaces, and to simultaneously
describe families of vectors whose shifts form frames for the SI spaces
they generate.

Series: Analysis Seminar

In this seminar I will discuss current work, joint with AndrewVince and Alex Grant. The goal is to tie together several related areas, namelytiling theory, IFS theory, and NCG, in terms most familiar to fractal geometers.Our focus is on the underlying code space structure. Ideas and a conjecture willbe illustrated using the Golden b tilings of Robert Ammann

Series: Analysis Seminar

A well-known elementary linear algebra fact says that any linear
independent set of vectors in a finite-dimensional vector space cannot
have more elements than any spanning set. One way to obtain an analog of
this result in the infinite
dimensional setting is by replacing the comparison of cardinalities
with a more suitable concept - which is the concept of densities.
Basically one needs to compare the cardinalities locally everywhere and
then take the appropriate limits. We provide a rigorous
way to do this and obtain a universal density theorem that generalizes
many classical density results. I will also discuss the connection
between this result and the uncertainty principle in harmonic analysis.

Series: Analysis Seminar

Finding and understanding patterns in data sets is of significant
importance in many applications. One example of a simple pattern is the
distance between data points, which can be thought of as a 2-point
configuration. Two classic questions, the Erdos distinct
distance problem, which asks about the least number of distinct
distances determined by N points in the plane, and its continuous
analog, the Falconer distance problem, explore that simple pattern.
Questions similar to the Erdos distinct distance problem and
the Falconer distance problem can also be posed for more complicated
patterns such as triangles, which can be viewed as 3-point
configurations. In this talk I will present recent progress on Falconer
type problems for simplices. The main techniques used come
from analysis and geometric measure theory.