Let denote the set of all algebraic polynomials of degree at most n with real coefficients. Associated with a set of poles we define the rational function spaces
Associated with a set of poles , we define the rational function spacesIt is an interesting problem to characterize sets for which is not dense in , where denotes the space of all continuous functions equipped with the uniform norm on . Akhieser showed that the density of is characterized by the divergence of the series .
In this paper, we show that the so-called Clarkson–Erdős–Schwartz phenomenon occurs in the non-dense case. Namely, if is not dense in , then it is “very much not so”. More precisely, we prove the following result.
Theorem
Let . Suppose is not dense in , that is,Then every function in the uniform closure of in can be extended analytically throughout the set .