Vol. 27, No. 1, 1968

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Uniform approximation by polynomials with integral coefficients I

Le Baron O. Ferguson

Vol. 27 (1968), No. 1, 53–69

Throughout this paper A will denote a discrete subring of the complex number plane C with rank 2. For example, A could be the Gaussian integers Z + iZ, where Z denotes the rational integers, or the ring of integers of any imaginary quadratic field. We are concerned with characterizing those functions defined on a compact subset X of C which can be uniformly approximated by polynomials with coefficients in A. We say that such functions are A-approximable on X. We also consider the real case where X is any compact subset of the reals R and the coefficients of the approximating polynomials lie in Z or any discrete subring of R. The real case is completely solved in the sense that a necessary and sufficient condition in order that a function can be so approximated is found. The complex case is solved if, in addition to being compact, X either has transfinite diameter at least unity or void interior and connected complement.

Mathematical Subject Classification
Primary: 41.15
Received: 27 September 1966
Revised: 16 August 1967
Published: 1 October 1968
Le Baron O. Ferguson