Padé approximants, orthogonal polynomials, and $S$-curves

The problem of efficient analytic continuation (summation) of a given power series outside its circle of convergence is a classical problem of complex analysis. In the talk we suppose to give a review of some methods of investigation of this problem, based on the use of diagonal Padé approximants and some their generalizations.
The main class of the functions in question is the class of multivalued analytic functions with a finite number of branch points in the complex plane. In this class of functions the denominators of generalized Padé approximants are the non-Hermitian orthogonal polynomials with respect to variable (depending on the degree of polynomial) weight function. The distribution of the zeros of these orthogonal polynomials may be characterized in terms of an extremal theoretical potential problem considered in some class of compact sets, ‘admissible’ for a given multi-valued function. Such extremal compact set is unique, consists of a finite number of analytic arcs (closures of critical trajectories of a quadratic differential) and it is characterized by some property of simmetry (so-called $S$-property). The limit distribution of the zeros of the denominators of the Padé approximants coincides with the equilibrium measure for that $S$-curve. The initial power series continues into the complement to the extremal compact set as a holomorphic (i.e., single-valued analytic) function. The diagonal Padé approximants converge in logarithmic capacity at a geometric rate to this holomorphic continuation of the original function. Based on the known distribution of the zeros and poles of Padé approximants, it is possible to solve the problem of uniform approximation of the original function using Padé approximants.