The uniform closure of non-dense rational spaces on the unit interval

AbstractLet 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}⊂R⧹[-1,1] we define the rational function spaces Pn(a1,a2,…,an):=f:f(x)=b0+∑j=1nbjx-aj,b0,b1,…,bn∈R.Associated with a set of poles {a1,a2,…}⊂R⧹[-1,1], we define the rational function spacesP(a1,a2,…):=⋃n=1∞Pn(a1,a2,…,an).It is an interesting problem to characterize sets {a1,a2,…}⊂R⧹[-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 characterized by the divergence of the series ∑n=1∞an2-1.In this paper, we show that the so-called Clarkson–Erdős–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.TheoremLet {a1,a2,…}⊂R⧹[-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 C⧹{-1,1,a1,a2,…}