Title: Padé approximation and Jacobi matrices
Speaker: Maxim Derevyagin
Abstract. Recall that a Padé approximant is a rational function whose Maclaurin expansion agrees with the given power series centered at 0 up to a certain order. Padé approximation is extremely important in numerical analysis and applied sciences since in many situations Padé approximants provide us with a very fast approximation scheme and, more importantly, they contain information about singularities of the underlying function unlike the polynomial approximations. Clearly, the scheme can be applied when it is possible to prove convergence. So, at first we will discuss the issue of convergence of Padé approximants and then translate the problem into the operator language. More precisely, it will be shown that Padé approximants are related to continued fractions, orthogonal polynomials, and Jacobi matrices. After that, some results on the locally uniform convergence of Padé approximants will be presented and, as a motivation for future research in this direction, the Bessis-Perotti idea of extracting information from highly noisy data will be reviewed.
Besides, we will also consider some generalizations of the scheme and see how Padé approximation gives rise to classical numerical algorithms, discrete and continuous integrable systems, biorthogonal rational functions, Nevanlinna-Pick interpolation, and generalized eigenvalue problems.