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Series: Analysis Seminar

We impose standard $T1$-type assumptions on a Calderón-Zygmund operator $T$, and deduce that for bounded compactly supported functions $f,g$ there is a sparse bilinear form $\Lambda$ so that
$$
|\langle T f, g \rangle | \lesssim \Lambda (f,g).
$$
The proof is short and elementary. The sparse bound quickly implies all the standard mapping properties of a Calderón-Zygmund on a (weighted) $L^p$ space.

Series: Analysis Seminar

For a compact subset $A$ of $R^n$ , let $A(k)$ be the Minkowski sum of $k$ copies of $A$, scaled by $1/k$. It is well known that $A(k)$ approaches the convex hull of $A$ in Hausdorff distance as $k$ goes to infinity. A few years ago, Bobkov, Madiman and Wang conjectured that the volume of $A(k)$ is non-decreasing in $k$, or in other words, that when the volume deficit between the convex hull of $A$ and $A(k)$ goes to $0$, it actually does so monotonically. While this conjecture holds true in dimension $1$, we show that it fails in dimension $12$ or greater. Then we consider whether one can have monotonicity of convergence of $A(k)$ when its non-convexity is measured in alternate ways. Our main positive result is that Schneider’s index of non-convexity of $A(k)$ converges monotonically to $0$ as $k$ increases; even the convergence does not seem to have been known before. We also obtain some results for the Hausdorff distance to the convex hull, along the way clarifying various properties of these notions of non-convexity that may be of independent interest.Joint work with Mokshay Madiman, Matthieu Fradelizi and Arnaud Marsiglietti.

Series: Analysis Seminar

Given
a set S of positive measure on the unit circle, a set of integers K is
an interpolation set (IS) for S if for any data {c(k)} in l^2(K) there
exists a function f in L^2(S) such that its Fourier coefficients satisfy
f^(k)=c(k) for all k in K. In
the talk I will discuss the relationship between the concept of IS and
the existence of arbitrarily long arithmetic progressions with specified
lengths and step sizes in K. Multidimensional analogues of this subject
will also be considered.This talk is based on joint work with Alexander Olevskii.

Series: Analysis Seminar

The theory of two variable orthogonal polynomials is not very well
developed. I will discuss some recent results on two variable orthogonal
polynomials on the bicircle and time permitting on the square associate
with orthogonality measures that are one over a trigonometric
polynomial. Such measures have come to be called Bernstein-Szego
measures.
This is joint work with Plamen Iliev and Greg Knese.

Series: Analysis Seminar

We show that multilinear dyadic paraproducts and Haar multipliers, as well as their commutators with locally integrable functions, can be pointwise dominated by multilinear sparse operators. These results lead to various quantitative weighted norm inequalities for these operators. In particular, we introduce multilinear analog of Bloom's inequality, and prove it for the commutators of the multilinear Haar multipliers.

Series: Analysis Seminar

The Linear Independence of Time-Frequency Translates Conjecture,
also known as the HRT conjecture, states that any finite set of
time-frequency translates of a given $L^2$ function must be linearly
independent. This conjecture, which was first stated in print in 1996,
remains open today. We will discuss this conjecture, its relation to
the Zero Divisor Conjecture in abstract algebra, and the (frustratingly
few) partial results that are currently available.

Series: Analysis Seminar

We show that the multiwavelets, introduced by Alpert in 1993, are
related to type I Legendre-Angelesco multiple orthogonal polynomials. We give explicit formulas for these Legendre-Angelesco polynomials and
for the Alpert multiwavelets. The multiresolution analysis can be done entirely using Legendre polynomials, and we give an algorithm,
using Cholesky factorization, to compute the multiwavelets and a method, using the Jacobi matrix for Legendre polynomials, to compute the
matrices in the scaling relation for any size of the multiplicity of the
multiwavelets.Based on joint work with J.S. Geronimo and P. Iliev

Series: Analysis Seminar

I will present a discrete family of multiple orthogonal polynomials defined
by a set of orthogonality conditions over a non-uniform lattice with
respect to different q-analogues of Pascal distributions. I will obtain
some algebraic properties for these polynomials (q-difference equation and
recurrence relation, among others) aimed to discuss a connection with an
infinite Lie algebra realized in terms of the creation and annihilation
operators for a collection of independent ascillators. Moreover, if time
allows, some vector equilibrium problem with constraint for the nth root
asymptotics of these multiple orthogonal polynomials will be discussed.

Series: Analysis Seminar

Series: Analysis Seminar

An
equiangular tight frame (ETF) is a set of unit vectors whose coherence
achieves the Welch bound. Though they arise in many applications, there
are only a
few known methods for constructing ETFs. One of the most popular
classes of ETFs, called harmonic ETFs, is constructed using the
structure of finite abelian groups. In this talk we will discuss a broad
generalization of harmonic ETFs. This generalization allows
us to construct ETFs using many different structures in the place of
abelian groups, including nonabelian groups, Gelfand pairs of finite
groups, and more. We apply this theory to construct an infinite family
of ETFs using the group schemes associated with
certain Suzuki 2-groups. Notably, this is the first known infinite
family of equiangular lines arising from nonabelian groups.