On the closure of a system of polynomials in a weighted space (Q1362404)

Let \[ E= \Biggl\{z:\text{Re }z= 0,\;\text{Im }z={\pi\over 2}\Biggr\}\cup \Biggl\{z:\text{Re }z=0,\;\text{Im }z=-{\pi\over 2}\Biggr\} \] and let \(w(t)\geq 1\) be a real measurable function on \(E\) such that \(t^nw^{-1}(t)\to 0\), \(|t|\to +\infty\), \(t\in E\), \(n=0,1,\dots\)\ . The author considers the space \(\mathbb{C}_w(E)\) of continuous functions on \(E\) satisfying the condition \(f(t)w^{-1}(t)\to 0\), \(|t|\to+\infty\), \(t\in E\) and establishes the integral criterion for the completeness of polynomials in \(\mathbb{C}_w(E)\). Moreover, he had described the closure of a system of polynomials in metric of \(\mathbb{C}_w(E)\) when this closure does not equal to the whole of the space.