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

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} ⊂ R\ [−1, 1] we define the rational function spaces Pn(a1, a2, . . . , an) := 8< : : f(x) = b0 + n X j=1 bj x− aj , b0, b1, . . . , bn ∈ R 9= ; . Associated with a set of poles {a1, a2, . . . } ⊂ R\[−1, 1] , we define the rational function spaces P(a1, a2, . . . ) := ∞ [ n=1 Pn(a1, a2, . . . , an) . It is an interesting problem to charcterize 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 P∞ n=1 p an − 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. Theorem. Let {a1, a2, . . . } ⊂ R \ [−1, 1]. Suppose P(a1, a2, . . . ) is not dense in C[−1, 1], that is, ∞ X