Rational and Polynomial Interpolation of Analytic Functions with Restricted Growth Original Research Article

Let f be an analytic function on a domain D⊂C∪{∞} and rn the rational function of degree n with poles at the points Bn={bni}ni=1, interpolating to f at the points An={ani}ni=0⊂D. A fundamental question is whether it is possible to choose the points An and Bn so that rn converges locally uniformly to f on D for every analytic function f on D. In some situations the interpolation points must be allowed to approach the boundary of D as n tends to infinity and then we cannot obtain convergence for every analytic f on D. If we restrict the growth of f(z) when z goes to the boundary of D, we still have some positive convergence results that we prove here.