Vol. 26, No. 2, 1968

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Uniform approximation by polynomials with integral coefficients. II

Le Baron O. Ferguson

Vol. 26 (1968), No. 2, 273–281

Let A be a discrete subring of C of rank 2. Let X be a compact subset of C with transfinite diameter not less than unity or with transfinite diameter less than unity, void interior, and connected complement. In an earlier paper we characterized the complex valued functions on X which can be uniformly approximated by elements from the ring of polynomials A[z]. In this paper the same problem is studied where X is a compact subset of C with transfinite diameter d(X) less than unity and with nonvoid interior. It is also studied for certain compact subsets of Cn where n is any positive integer. These subsets will have the property that every continuous function holomorphic on the interior is uniformly approximable by complex polynomials. A large class of sets of this type is shown to exist.

Mathematical Subject Classification
Primary: 41.15
Received: 17 November 1966
Revised: 16 August 1967
Published: 1 August 1968
Le Baron O. Ferguson