Chebyshev–Padé approximants for multivalued functions

The paper discusses the connection between the linear Chebyshev–Padé approximants for an analytic function $f$ and diagonal type I Hermite–Padé polynomials for the set of functions $[1, f_1, f_2]$, where the pair of functions $f_1$, $f_2$ forms a Nikishin system. Both problems can ultimately be reduced to certain convergence problems for multipoint Padé approximants. On the other hand, the denominators of multipoint Padé approximants are non-Hermitian orthogonal polynomials with analytical weights. Thus, to study all the above problems, the general method created by Herbert Stahl can be applied. Stahl’s method is not yet sufficiently developed to obtain general results on these problems. In particular, many key convergence problems for Chebyshev–Padé approximants for functions with arbitrary configurations of branch points remain open. In this paper, we consider several important general and particular results related to this case, some already well known, and also formulate two general hypotheses in the indicated direction.