Ahmad Barhoumi: Padé approximants for functions with four branch points

Abstract:

Padé approximants are rational functions which interpolate a power series at its center to the "highest possible" order. They can be constructed in an algorithmic manner; their coefficients satisfy linear equations whose coefficients are moments of the approximated function. The location of the poles is not fixed apriori, but for many reasonable examples the poles behave in a structured manner. This can be explained, for large classes of approximated functions, by exploiting a connection with non-Hermitian orthogonal polynomials.

In this talk, I will discuss the connection with orthogonal polynomials and explain the behavior of the poles and convergence properties of Padé approximants while focusing on functions with four branch points as a running example. I will end with some of my work, joint with Maxim Yattselev, on the approximation of non-generic functions.