A Montessus de Ballore Theorem for M ultivariate Pad6 Approximants

During the last few years several authors have tried to generalize the concept of Pad& 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. Sot. London Ser. A 342 (1975), 341-372), J. Karlsson and H. Wallin (“Pad& 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 Cp. This problem is discussed in Section 3. A definition of multivariate Pade approximant, which was introduced by A. A. M. Cuyt (“Padi: Approximants for Operators: Theory and Applications,” Lecture Notes in Mathematics No. 1065, SpringerVerlag, Berlin, 1984; J. Mad 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 Pad& approximants: covariance properties, block-structure of the Pad&table, the e-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.