Seminars and Colloquia by Series

On the uniqueness sets in the Bergmann-Fock space

Series
Analysis Seminar
Time
Wednesday, April 13, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mishko MitkovskiSchool of Mathematics, Georgia Tech
It is well known that, via the Bargmann transform, the completeness problems for both Gabor systems in signal processing and coherent states in quantum mechanics are equivalent to the uniqueness set problem in the Bargmann-Fock space. We introduce an analog of the Beurling-Malliavin density to try to characterize these uniqueness sets and show that all sets with such density strictly less than one cannot be uniqueness sets. This is joint work with Brett Wick.

Orthogonal Rational Functions and Rational Gauss-type Quadrature Rules

Series
Analysis Seminar
Time
Wednesday, April 6, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Karl DeckersGeorgia Tech
Consider a positive bounded Borel measure \mu with infinite supporton an interval [a,b], where -oo <= a < b <= +oo, and assume we have m distinctnodes fixed in advance anywhere on [a,b]. We then study the existence andconstruction of n-th rational Gauss-type quadrature formulas (0 <= m <= 2)that approximate int_{[a,b]} f d\mu. These are quadrature formulas with npositive weights and n distinct nodes in [a,b], so that the quadratureformula is exact in a (2n - m)-dimensional space of rational functions witharbitrary complex poles fixed in advance outside [a,b].

Scattering for the cubic Klein Gordon equation in two space dimensions

Series
Analysis Seminar
Time
Wednesday, March 16, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Betsy StovallUCLA
We will discuss a proof that finite energy solutions to the defocusing cubicKlein Gordon equation scatter, and will discuss a related result in thefocusing case. (Don't worry, we will also explain what it means for asolution to a PDE to scatter.) This is joint work with Rowan Killip andMonica Visan.

Cantor Boundary Behavior of Analytic Functions

Series
Analysis Seminar
Time
Thursday, March 10, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ka-Sing LauHong Kong Chinese University
There is a large literature to study the behavior of the image curves f(\partial {\mathbb D}) of analytic functions f on the unit disc {\mathbb D}. Our interest is on the class of analytic functions f for which the image curves f(\partial {\mathbb D}) form infinitely many (fractal) loops. We formulated this as the Cantor boundary behavior (CBB). We develop a general theory of this property in connection with the analytic topology, the distribution of the zeros of f'(z) and the mean growth rate of f'(z) near the boundary. Among the many examples, we showed that the lacunary series such as the complex Weierstrass functions have the CBB, also the Cauchy transform F(z) of the canonical Hausdorff measure on the Sierspinski gasket, which is the original motivation of this investigation raised by Strichartz.

Energy estimates for the random displacement model

Series
Analysis Seminar
Time
Wednesday, March 9, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael LossSchool of Mathematics, Georgia Tech
This talk is about a random Schroedinger operator describing the dynamics of an electron in a randomly deformed lattice. The periodic displacement configurations which minimize the bottom of the spectrum are characterized. This leads to an amusing problem about minimizing eigenvalues of a Neumann Schroedinger operator with respect to the position of the potential. While this configuration is essentially unique for dimension greater than one, there are infinitely many different minimizing configurations in the one-dimensional case. This is joint work with Jeff Baker, Frederic Klopp, Shu Nakamura and Guenter Stolz.

Beyond Calderon's algebra

Series
Analysis Seminar
Time
Wednesday, March 2, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Camil MuscaluCornell
Calderon's algebra can be thought of as a world whichincludes singular integral operators and operators of multiplicationwith functions which grow at most linearly (more precisely, whose firstderivatives are bounded).The goal of the talk is to address and discuss in detail the followingnatural question: "Can one meaningfully extend it to include operatorsof multiplication with functions having polynomial growth as well ?".

Regularity of Solutions to Extremal Problems in Bergman Spaces

Series
Analysis Seminar
Time
Wednesday, February 16, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tim FergusonUniversity of Michigan
I will discuss linear extremal problems in the Bergman spaces $A^p$ ofthe unit disc and a theorem of Ryabykh about regularity of thesolutions to these problems. I will also discuss extensions I havefound of Ryabykh's theorem in the case where $p$ is an even integer.The proofs of these extensions involve Littlewood-Paley theory and abasic characterization of extremal functions.

Are the Degrees of Best (Co)Convex and Unconstrained Polynomial Approximation the Same?

Series
Analysis Seminar
Time
Wednesday, January 26, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Dany Leviatan Tel Aviv University
Let C[-1, 1] be the space of continuous functions on [-1, 1], and denote by \Delta^2 the set of convex functions f \in C[-1, 1]. Also, let E_n(f) and En^{(2)}_n(f) denote the degrees of best unconstrained and convex approximation of f \in \Delta^2 by algebraic polynomials of degree < n, respectively. Clearly, E_n(f) \le E^{(2)}_n (f), and Lorentz and Zeller proved that the opposite inequality E^{(2)}_n(f) \le CE_n(f) is invalid even with the constant C = C(f) which depends on the function f \in \Delta^2. We prove, for every \alpha > 0 and function f \in \Delta^2, that sup{n^\alpha E^{(2)}_n(f) : n \ge 1} \le c(\alpha)sup{n^\alpha E_n(f): n \ge 1}, where c(\alpha) is a constant depending only on \alpha. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (-1,1) is also investigated. It turns out that there are substantial differences between the cases s \le 1 and s \ge 2.

Pages