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

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Abstract

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):=f:f(x)=b0+j=1nbjx-aj,b0,b1,,bnR.

Associated with a set of poles {a1,a2,}R[-1,1], we define the rational function spacesP(a1,a2,):=n=1Pn(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=1an2-1.

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,n=1an2-1<.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,}.

MSC

primary 41A17
secondary 26D10
26D15

Keywords

Bernstein-type inequalities
Rational functions
Density
Uniform closure

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Research is supported, in part, by NSF under Grant No. DMS-0070826