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

[摘要] Let P-n denote the set of all algebraic polynomials of degree at most n with real coefficients. Associated with a set of poles {a(1), a(2),..., a(n)} subset of R\[-1, 1] we define the rational function spaces P-n(a(1), a(2),...,a(n)) : = {f : f(x) = b(0) + Sigma(j=1)(11) bj/x - a(j), b(0), b(1),...,b(n) is an element of R}. Associated with a set of poles {a(1), a(2),...} subset of R\[-1, 1], we define the rational function spaces P-n(a(1), a(2),...) : =boolean ORn=1infinity P-n(a(1), a(2),..., a(n)). It is an interesting problem to characterize sets {a(1), a(2),...} subset of R\[-1, 1] for which P(a(1), a(2),...) 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(a(1), a(2),...) is characterized by the divergence of the series Sigma(n=1)(infinity)roota(n)(2) - 1. In this paper, we show that the so-called Clarkson-Erdos-Schwartz phenomenon occurs in the non-dense case. Namely, if P(a(1), a(2),...) is not dense in C[-1, 1], then it is very much not so. More precisely, we prove the following result.

[发布日期] 2004-12-01

[关键词] Bernstein-type inequalities;rational functions;density;uniform closure