Georgia Institute of Technology College of Sciences

The asymptotic analysis of orthogonal polynomials with respect to a varying weight has found many interesting applications in approximation theory, random matrix theory and other areas. It has also stimulated a further development of the logarithmic potential theory, since the equilibrium measure in an external field associated with these weights enters the leading term of the asymptotics and its support is typically the place where zeros accumulate and oscillations occur. In a rather broad class of problems the varying weight on the real

One of the basic problems of Harmonic analysis is to determine ifa given collection of functions is complete in a given Hilbert space. Aclassical theorem by Beurling and Malliavin solved such a problem in thecase when the space is $L^2$ on an interval and the collection consists ofcomplex exponentials.

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.

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.

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.

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

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.

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.

In this talk we will connect several different areas of mathematical analysis: complex analysis, harmonic analysis and functiontheory all in the hopes of gaining a better understanding of Carleson measures for certain classes of function spaces.