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Series: Analysis Seminar

We discuss a global weighted estimate for a class of divergence form elliptic operators with BMO coefficients on Reifenbergflat domains. Such an estimate implies new global regularity results in Morrey, Lorentz, and H\"older spaces for solutionsof certain nonlinear elliptic equations. Moreover, it can also be used to obtain a capacitary estimate to treat a measuredatum quasilinear Riccati type equations with nonstandard growth in the gradient.

Series: Analysis Seminar

We consider the influence of an incompressible drift on the expected exit time of a diffusing particle from a bounded domain. Mixing resulting from an incompressible drift typically enhances diffusion so one might think it always decreases the expected exit time. Nevertheless, we show that in two dimensions, the only simply connected domains for which the expected exit time is maximized by zero drift are the discs.

Series: Analysis Seminar

The Schur-Agler class is a subclass of the bounded analytic functions on the polydisk with close ties to operator theory. We shall describe our recent investigations into the properties of rational inner functions in this class. Non-minimality of transfer function realization, necessary and sufficient conditions for membership (in special cases), and low degree examples are among the topics we will discuss.

Series: Analysis Seminar

A separated sequence of real numbers is called a Polya sequence if the only entire functions of zero type which are bounded on this sequence are the constants. The Polya-Levinson problem asks for a description of all Polya sequences. In this talk, I will present some points of the recently obtained solution. The approach is based on the use of Toeplitz operators and de Branges spaces of entire functions. I will also present some partial results about the related Beurling gap problem.

Series: Analysis Seminar

Sobolev orthogonal polynomials in two variables are defined via
inner products involving gradients. Such a kind of inner product appears
in connection with several physical and technical problems. Matrix
second-order partial differential equations satisfied by Sobolev
orthogonal polynomials are studied. In particular, we explore the
connection between the coefficients of the second-order partial
differential operator and the moment functionals defining the Sobolev
inner product. Finally, some old and new examples are given.

Series: Analysis Seminar

The asymptotic theory is developed for polynomial sequences that
are generated by the three-term higher-order recurrence
Q_{n+1} = zQ_n - a_{n-p+1}Q_{n-p}, p \in \mathbb{N}, n\geq p,
where z is a complex variable and the coefficients a_k are
positive and satisfy the perturbation condition \sum_{n=1}^\infty
|a_n-a|<\infty . Our results generalize known results for p = 1,
that is, for orthogonal polynomial sequences on the real line that
belong to the Blumenthal-Nevai class. As is known, for p\geq 2,
the role of the interval is replaced by a starlike set S of
p+1 rays emanating from the origin on which the Q_n satisfy a
multiple orthogonality condition involving p measures. Here we
obtain strong asymptotics for the Q_n in the complex plane
outside the common support of these measures as well as on the
(finite) open rays of their support. In so doing, we obtain an
extension of Weyl's famous theorem dealing with compact
perturbations of bounded self-adjoint operators. Furthermore, we
derive generalizations of the classical Szeg\"o functions, and
we show that there is an underlying Nikishin system hierarchy for
the orthogonality measures that is related to the Weyl functions.
Our results also have application to Hermite-Pad\'e approximants
as well as to vector continued fractions.

Series: Analysis Seminar

Low discrepancy point distributions play an important role in many
applications that require numerical integration. The methods of
harmonic analysis are often used to produce new or de-randomize known
probabilistic constructions. We discuss some recent results in this
direction.

Series: Analysis Seminar

Let A and B be attractors of two point-fibred iterated function
systems with coding maps f and g. A transformations from A into B can be
constructed by composing a branch of the inverse of f with g. I will outline
the shape of the theory of such transformations, which are termed "fractal"
because their graphs are typically of non-integer dimension. I will also
describe the remarkable geometry of these transformations when the
generating iterated functions systems are projective. Finally, I will show how they
can be used to provide new insights into dynamical systems and also how
they can be used to manipulate, filter, process and efficiently store digital
images, and how they can be used in image synthesis, leading to
applications in the visual arts.

Series: Analysis Seminar

We prove an extension of the Wiener inversion theorem for
convolution of summable series, allowing the terms to take values in a
space of bounded linear operators. The resulting algebra is no longer
commutative due to the composition of operators. Inversion theorems
arise naturally in the context of proving dispersive estimates for the
Schr\"odinger and wave equation and lead to scale-invariant conditions
for the class of admissible potentials.
All results are joint work with Marius Beceanu.

Series: Analysis Seminar

A2 conjecture asked to have a linear estimate for simplest weighted singular operators in terms of the measure of goodness of the weight in question.We will show how the paradigm of non-homogeneous Harmonic Analysis (and especially its brainchild, the randomized BCR) was used to eventually solve this conjecture.