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An analogue of Fabry's theorem for generalized Padé approximants

The current theory of Padé approximation emphasises results of an inverse character, when conclusions about the properties of the approximated function are drawn from information about the behaviour of the approximants. In this paper Gonchar's conjecture is proved; it states that analogues of Fabry's classical `ratio' theorem hold for rows of the table of Padé approximants for orthogonal expansions, multipoint Padé approximants and Padé-Faber approximants. These are the most natural generalizations of the construction of classical Padé approximants. For these Gonchar's conjecture has already been proved by Suetin. The proof presented here is based, on the one hand, on Suetin's result and, on the other hand, on an extension of Poincaré's theorem on recurrence relations with coefficients constant in the limit, which is obtained in the paper. Bibliography: 19 titles.