Colloquium: "Rational approximants to holomorphic functions of one variable"
Abstract: Rational approximation to analytic functions of one complex variable, on compact subsets of their domain of holomorphy, is a standard topic in classical analysis with applications to number theory, spectral theory and numerical analysis for example; over the last thirty years or so, it also became an important tool in engineering to identify and design various systems in the frequency domain. In this talk we discuss the speed of approximation as the degree $n$ of the approximant goes large, in particular the Gonchar conjecture bounding from above the liminf of the $n$-th root of the error, as well as its proof by Parfenov (Prokhorov in the multiply connected case). The sharpness of this upper bound had been shown previously by Gonchar and Rakhmanov, using interpolation and potential-theoretic techniques initiated by J. Nuttall and H. Stahl. Dwelling on joint work with H. Stahl and M. Yattselev, we further explain why branches of functions that live on a different Riemann surface than the sphere meet Gonchar's bound, and how this causes the normalized counting measure of the poles of their rational approximants (optimal and near-optimal) to converge (weak star) to a certain distribution solving an extremal problem from (logarithmic) potential theory. This sheds light on how the singularities of the approximants reflect those of the approximated function.
Host: Brett Wick
Tea will be served at 3:30pm in room 200.