A Montessus de Ballore theorem for multivariate Padé approximants

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During the last few years several authors have tried to generalize the concept of Padé approximant to multivariate functions and to prove a generalization of Montessus de Ballore's theorem. We refer, e.g., to J. Chisholm and P. Graves-Morris ( Proc. Roy. Soc. London Ser. A 342 (1975) , 341–372), J. Karlsson and H. Wallin (“Padé and Rational Approximations and Applications” (E. B. Saff and R. S. Varga, Eds.), pp. 83–100, Academic Press, 1977, C. H. Lutterodt ( J. Phys. A 7, No. 9 (1974), 1027–1037; J. Math. Anal. Appl. 53 (1976), 89–98; preprint, Dept. of Mathematics, University of South Florida, Tampa, Florida, 1981 ). However, it is a very delicate matter to generalize Montessus de Ballore's result from C to C p . This problem is discussed in Section 3. A definition of multivariate Padé approximant, which was introduced by A. A. M. Cuyt (“Padé Approximants for Operators: Theory and Applications,” Lecture Notes in Mathematics No. 1065, Springer-Verlag, Berlin, 1984; J. Math. Anal. Appl. 96 (1983), 283–293) and which is repeated in Section 1, is a generalization that allows one to preserve many of the properties of the univariate Padé approximants: covariance properties, block-structure of the Padé-table, the ε-algorithm, the qd-algorithm, and so on. It also allows one to formulate a Montessus de Ballore theorem, which is presented in Section 2; up to now it is probably the most “Montessus de Ballore”-like version existing for the multivariate case. Illustrative numerical results are given in Section 4.