Some approximation theorems

Abstract

The general theme of this note is illustrated by the following theorem:Theorem 1. Suppose K is a compact set in the complex plane and 0belongs to the boundary ∂K. Let A(K) denote the space of all functions f on K such that f is holo morphic in a neighborhood of K and f(0) = 0.Also for any givenpositive integer m, let A(m, K) denote the space of all f such that f is holomorphic in a neighborhood of K and f(0) =f′(0) = ... =f ^(m)(0) = 0.Then A(m, K) is dense in A(K) under the supre mum norm on K provided that there exists a sector W = re ^iθ; 0≤r≤ δ,α≤ θ≤ β such that W ∩ K = 0. (This is the well- known Poincare’s external cone condition). We present various generalizations of this result in the context of higher dimensions replacing holomorphic with harmonic.