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The convergence of diagonal Padé approximants is investigated. Starting with a review of general results that are connected with convergence in capacity the investigation then concentrates on a detailed study of diagonal Padé approximants to hyperelliptic functions. Special emphasis is given to the study of spurious poles. Locally uniform convergence is proved after clearing diagonal Padé approximants from spurious poles, and it is shown that under certain conditions the Baker-GammelWills conjecture holds true for hyperelliptic functions. MOTS-CLÉS : Approximation rationnelle, approximation de Padé, spurious (faux) pôles, fonctions hyperelliptiques, la conjecture de BakerGammel-Wills. KEY-WORDS : Rational approximation, Padé approximation, spurious poles, hyperelliptic functions, Baker-Gammel-Wills conjecture. AMS Classification : 41A21, 41A25, 30E10. Annales de la Faculté des Sciences de Toulouse n° spécial Stieltjes, 1996 (1) TFH-Berlin/FB 2; Luxemburgerstr. 10; 13353 Berlin: Germany. Research supported by the Deutsche Forschungsgemeinschaft (AZ: Sta 299/8-1). The paper was written while the author was visiting the Steklov Institute, Moskow.

Diagonal Padé approximants to hyperelliptic functions