Asymptotically Neutral Distributions of Electrons and Polynomial Approximation

Let D be a bounded simply connected region (connected open set) in the complex plane. We wish to approximate holomorphic functions in D by polynomials; the error must become uniformly small on every fixed closed subset of D. By Runge's theorem such approximation is always possible. However, we impose an additional condition: the zeros of the approximating polynomials must belong to a prescribed set P. Polynomials whose zeros lie in P will be referred to as F-polynomials. It will be assumed for simplicity that F and D are disjoint. P will be called a polynomial approximation set relative to D if and only if every zero free holomorphic function in D can be approximated by F-polynomials. It may be observed that, if F is a polynomial approximation set and J is a simple arc in D, then every continuous function on J can be uniformly approximated by F-polynomials. G. R. MacLane 4 has shown that every rectifiable Jordan curve is a polynomial approximation set relative to its interior. Recently M. D. Thompson 5 gave a new proof of this result, extending it to a different special class of Jordan curves. M.D. Thompson 5 and the author 3 have also considered certain unbounded sets F. In the present paper we determine all bounded sets F that can serve as polynomial approximation sets relative to D. The discussion will be based on the notion of asymptotically neutral families. A family of finite sequences