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Series: Analysis Seminar

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.

Series: Analysis Seminar

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].

Series: Analysis Seminar

We consider the 1d wave equation and prove the propagation of the wave provided that the potential is square summable on the half-line. This result is sharp.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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 ?".

Series: Analysis Seminar

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.

Series: Analysis Seminar

I will survey recent results about the convergence of the Wash-Fourier series near L1. Joint work with Michael Lacey.

Series: Analysis Seminar

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.