Tschebyshev-Padé approximations for multivalued functions

Abstract

We discuss the relation between the linear Tschebyshev-Padé approximations to analytic function f and the diagonal type I Hermite-Padé polynomials for the tuple of functions [1, f 1 , f 2 ] where the pair of functions f 1 , f 2 forms certain Nikishin system. An approach is proposed of how to extend the seminal Stahl's Theory for Padé approximations for multivalued analytic functions to the Tschebyshev-Padé approximations. The approach is based on the relation between Tschebyshev-Padé approximations and Hermite-Padé polynomials and also on a connection of Hermite-Padé polynomials and multipoint Padé approximants.Bibliography: [47] titles. Contents 1. Tschebyshev-Padé Approximations 1 2. Padé approximants at infinity 4 3. Hermite-Padé polynomials. 6 4. General interpolation problem 8 5. GRS theorem 10 6. Conjecture on Tschebyshev-Padé approximants 13 7. One numerical example 14 References 18