## Seminars and Colloquia by Series

### Polya sequences, gap theorems, and Toeplitz kernels

Series
Analysis Seminar
Time
Wednesday, October 20, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mishko MitkovskiGeorgia Tech
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.

### Sobolev orthogonal polynomials in two variables and partial differential equations

Series
Analysis Seminar
Time
Wednesday, October 6, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
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.

### Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

Series
Analysis Seminar
Time
Tuesday, October 5, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sasha AptekarevKeldish Institute for Applied Mathematics
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.

### Analysis in constructions of low discrepancy sets

Series
Analysis Seminar
Time
Wednesday, September 29, 2010 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dmitriy BilykUniversity of South Carolina
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.

### Theory and applications of fractal transformations

Series
Analysis Seminar
Time
Monday, September 27, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Michael BarnsleyDepartment of Mathematics, Australian National University
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.

### A non-commutative Wiener Inversion Theorem and Schroedinger dispersive estimates

Series
Analysis Seminar
Time
Wednesday, September 22, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Michael GoldbergUniversity of Cincinnati
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.

### Non-homogeneous Harmonic Analysis and randomized Beylkin--Coifman--Rokhlin algorithm (BCR): an application for the solutions of A2 conjecture.

Series
Analysis Seminar
Time
Wednesday, September 15, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Alexander VolbergMichigan State
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.

### The Point Mass Problem on the Real Line

Series
Analysis Seminar
Time
Wednesday, September 8, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Manwah WongGeorgia Tech
In this talk, I will talk about recent developments on the point mass problem on the real line. Starting from the point mass formula for orthogonal polynomials on the real line, I will present new methods employed to compute the asymptotic formulae for the orthogonal polynomials and how these formulae can be applied to solve the point mass problem when the recurrence coefficients are asymptotically identical. The technical difficulties involved in the computation will also be discussed.

### A Variational Estimate for Paraproducts

Series
Analysis Seminar
Time
Wednesday, September 1, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Yen DoGeorgia Tech
We show variational estimates for paraproducts, which can be viewed as bilinear generalizations of L\'epingle’s variational estimates for martingale averages or scaled families of convolution operators. The heart of the matter is the case of low variation exponents. Joint work with Camil Muscalu and Christoph Thiele.

### On complex orthogonal polynomials related with Gaussian quadrature of oscillatory integrals

Series
Analysis Seminar
Time
Wednesday, April 28, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Alfredo DeañoUniversidad Carlos III de Madrid (Spain)
We present results on the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. Our motivation comes from the fact that the zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral defined on the real axis and having a high order stationary point. The limit distribution of these zeros is also analyzed, and we show that they accumulate along a contour in the complex plane that has the S-property in the presence of an external field. Additionally, the strong asymptotics of the orthogonal polynomials is obtained by applying the nonlinear Deift--Zhou steepest descent method to the corresponding Riemann--Hilbert problem. This is joint work with D. Huybrechs and A. Kuijlaars, Katholieke Universiteit Leuven (Belgium).