A direct proof of Stahl's theorem for a generic class of algebraic functions

Abstract

Under the assumption that Stahl's $S$-compact set exists we give a short proof of the limiting distribution of the zeros of Padé polynomials and the convergence in capacity of diagonal Padé approximants for a generic class of algebraic functions. The proof is direct, rather than by contradiction as Stahl's original proof was. The ‘generic class’ means, in particular, that all the ramification points of the multisheeted Riemann surface of the algebraic function in question are of the second order (that is, all branch points of the function are of square root type). As a consequence, a conjecture of Gonchar relating to Padé approximations is proved for this class of algebraic functions. We do not use the relations of orthogonality for Padé polynomials. The proof is based on the maximum principle only. Bibliography: 19 titles.