Abstract

This paper shows that if f(z) is analytic in some neighborhood of the origin, but meromorphic in ℂn otherwise, with a denumerable non-accumulating pole sections in ℂn and if for each fixed ν the pole set of each (μ,ν) unisolvent rational approximant πμν(z) tends to infinity as μ′=mini≤n(μi)→∞, then f(z) must be entire in ℂn. This paper also shows a monotonicity property for the “error sequence” eμν=‖f(z)−πμν(z)‖K on compact subsets K of  ℂn.