Abstract:
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.
