Applied Numerical Mathematics 6 (1989/ 90) 341-360 North-Holland 341 THE CONVERGENCE OF PAD&TYPE APPROXIMANTS TO HOLOMORPHIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES Nicholas J. DARAS Jean Moreas 19, 152 32 Chalandri, Athens, Greece Our main purpose is to discuss the convergence of Pad&type approximants to a function which is holomorphic in an open subset 0 of C” (n > 1). If n = 1, a general method due to Eiermann gives the basic result [6]. This paper contains: (a) the study of some typical cases: 52 is an open polydisk centered at 0 and D = C”; (b) an answer to the question whether the approximation of a holomorphic function by a sequence of Pad&type approximants is possible; (c) four examples of Pad&type approximants converging to holomorphic functions, if n = 2. zyxwvutsrqponmlkjihgfe 0. Introduction and notations For a given summability method, the Okada theorem describes a domain in which an arbitrary power series can be analytically continued [7,10]. A generalized form of Okada’s result is a criterion for the efficiency of Pad&type approximants [6]. The first aim of this paper is to give the extension of this criterion in the case of an open polydisk in e; in doing so, we shall see that two versions of this extension can be derived. The techniques used are similar to those of Eiermann [ 61. The last two sections of this paper deal with an introduction to Pad&type approximants and some applications of the extension mentioned above. One section gives a brief account of definitions and properties of Pad&type approximants for formal double power series. The last section also demonstrates the applications of Theorems 1.3 and 1.4 to rational approximation. For an open polydisk in C n we shall use the notation A’( W; r) = A”( wl,. . . , w,; r,, . . . , r,) := {z=(q,..., Z,)Ec”:~Z,-wj~ <rj, j=l,2 )...) n}; the point w=(w,,..., w,,) is the center of the polydisk and r = (r,, . . . , r,,) is the polyradius of this polydisk. The closure of An( w; r) will be denoted by my>. If Ic2 is an open subset of C”, the set of all functions holomorphic in a will be denoted by 0( Sz). If j = 1, 2,. . . , n, then prj(1(2):= ([EC: g(z, ,..., zj_i, zj+i ,..., z,,) EQ=“-~ and (Zi> . . . . zj-1, s, zj+i ,..., zn) EQ). The boundary of a set D will be denoted by ao. Finally the function dist( -, - ) will be the Euclidean distance. 016%9274/ 90/ $03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)