The uniform closure of non-dense rational spaces on the unit interval
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Let Pn denote the set of all algebraic polynomials of degree at most n with real coefficients. Associated with a set of poles {a1, a2..., an} [- 1, 1] we define the rational function spaces Pn(a1, a2,..., an) := {f : f(x) = b0 + j=1n bj/x-aj, b0, b1,..., bn }. Associated with a set of poles {a1, a2,...}. [-1, 1], we define the rational function spaces P(a1, a22,...): =n=1 Pn(a1, a2,..., ann). It is an interesting problem to characterize sets {a1, a2,...}. [-1, 1] for which P(a1, a2,...) is not dense in C[-1,1], where C[-1, 1] denotes the space of all continuous functions equipped with the uniform norm on [-1, 1]. Akhieser showed that the density of P(a1, a2,...) is chracterized by the divergence of the series n=1 an2-1. In this paper, we show that the so-called Clarkson-Erdos-Schwartz phenomenon occurs in the non-dense case. Namely, if P(a1, a2,...) is not dense in C[-1, 1], then it is "very much not so". More precisely, we prove the following result. Theorem. Let {a1, a22,...} [-1, 1]. Suppose P(a1, a2,...) is not dense in C[-1, 1], that is, n=1 an2 - 1 < . Then every function in the uniform closure of P(a1, a2,...) in C[-1, 1] can be extended analytically throughout the set {-1, 1, a1, a22,...}. 2004 Elsevier Inc. All rights reserved.
