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Series: Analysis Seminar

The problem of weighted polynomial approximation of continuousfunctionson the real line was posted by S. Bernstein in 1924. It asks for adescription of theset of weights such that polynomials are dense in the space of continuousfunctions withrespect to the corresponding weighted uniform norm. Throughout the 20thcentury Bernstein's problem was studied by many prominent analysts includingAhkiezer, Carleson, Mergelyan andM. Riesz.In my talk I will discuss some of the complex analytic methods that can beapplied in Bernstein's problem along with a recently found solution.

Series: Analysis Seminar

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By the classical Weierstrass theorem, any function continuous on a compact set can be uniformly approximated by algebraic polynomials. In this talk we shall discuss possible
extensions of this basic result of analysis to approximation by homogeneous algebraic polynomials on
central symmetric convex bodies.
We shall also consider a related question of approximating convex bodies
by convex algebraic level surfaces. It has been known for some time time that any
convex body can be approximated arbirarily well by convex algebraic level surfaces. We
shall present in this talk some new results specifying rate of convergence.

Series: Analysis Seminar

Expansion in a wavelet basis provides useful information ona function in different positions and length-scales. The simplest example of wavelets are the Haar functions, which are just linearcombinations of characteristic functions of cubes, but often moresmoothness is preferred. It is well-known that the notion of Haarfunctions carries over to rather general abstract metric spaces. Whatabout more regular wavelets? It turns out that a neat construction canbe given, starting from averages of the indicator functions over arandom selection of the underlying cubes. This is yet anotherapplication of such probabilistic averaging methods in harmonicanalysis. The talk is based on joint work in progress with P. Auscher.

Series: Analysis Seminar

We consider ensembles of $N \times N$ Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law, as $N$ tends to infinity.

Series: Analysis Seminar

Series: Analysis Seminar

We consider boundedness of singular integrals in the two weight setting.
The problem consists in characterizing non-negative weights v and w
for which H: L^{p}(v)\mapsto L^{p}(w) for 1

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.