Rational interpolants with preassigned poles, theoretical aspects

Let f be an analytic function on a compact subset K of the complex plane C, and let r(n)(z) denote the rational function of degree n with poles at the points (b(ni))(i=1)(n) and interpolating f at the points (a(ni))(i=0)(n). We investigate how these points should be chosen to guarantee the convergen ce of r(n) to f as n --> infinity for all functions f analytic on K. When K has no "holes" (see [8] and [3]), it is possible to choose the poles (b(ni ))(i,n) without limit points on K. In this paper we study the case of gener al compact sets K, when such a separation is not always possible. This fact causes changes both in the results and in the methods of proofs. We consid er also the case of functions analytic in open domains. It turns out that i n our general setting there is no longer a "duality" ([8], Section 8.3, Cor ollary 2) between the poles and the interpolation points.