Rational interpolants with preassigned poles, theoretical aspects

Let ⨍ be an analytic function on a compact subset K of the complex plane ℂ, and let ${\displaystyle r_n\left(z\right)}$ denote the rational function of degree n with poles at the points ${\displaystyle {\left\{b_{ni}\right\}}^n\_\{i=1\}}$ and interpolating ⨍ at the points ${\displaystyle {\left\{a_{ni}\right\}}^n\_\{i=0\}}$. We investigate how these points should be chosen to guarantee the convergence of ${\displaystyle r_n}$ to ⨍ as n → ∞ for all functions ⨍ analytic on K. When K has no "holes" (see [8] and [3]), it is possible to choose the poles ${\displaystyle {\left\{b_{ni}\right\}}_{i,n}}$ without limit points on K. In this paper we study the case of general 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 consider also the case of functions analytic in open domains. It turns out that in our general setting there is no longer a "duality" ([8], Section 8.3, Corollary 2) between the poles and the interpolation points.