Title: Asymptotics of Chebyshev rational functions with respect to subsets of the real line

Abstract: Chebyshev and residual polynomials and their asymptotic behavior have been studied extensively. Subject to the constraint that their $L^\infty$ norm be bounded above by 1 on a given set, Chebyshev polynomials maximize the leading coefficient, while residual polynomials maximize the point evaluation at a given point. In this talk, I will describe some recent work on Chebyshev and residual extremal problems for rational functions with real poles with the constraint satisfied on subsets of $\overline{\R}$. In particular, I will describe a proof of root asymptotics under fairly general assumptions on the sequence of poles, as well as a proof of Szegő--Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau--Widom and DCT conditions. This is based on a preprint which is joint work with Benjamin Eichinger and Milivoje Lukić.

About the speaker: Giorgio Young is a fourth year PhD. student at Rice University, where his advisor is Milivoje Lukić. His research interests are the spectral theory of Schrödinger operators; orthogonal polynomials and Jacobi matrices; and the KdV equation.