Rational and polynomial interpolation of analytic functions with restricted growth

Let f be an analytic function on a domain D subset of C boolean OR {infinit y} and r(n) the rational function of degree n with poles at the points B-n = {b(ni)}(i = 1)(n), interpolating to f at the points A(n) = {a(ni)}(i = 0) (n) subset of D. A fundamental question is whether it is possible to choose the points A(n) and B-n so that r(n) converges locally uniformly to f on D for every analytic function f on D. In some situations the interpolation p oints 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 rest rict the growth of f(z) when z goes to the boundary of D, we still have som e positive convergence results that we prove here. (C) 2001 Academic Press.