Result: On the Denseness of Rational Systems: On the denseness of rational systems

Let \(\{a_k\}^n_{k=1} \subset \mathbb C \setminus [-1, 1]\) and \(\{c_k\}^n_{k=1}, |c_k|< 1\) be the numbers such that \[ a_k := \frac{c_k + c_k^{-1}}{2}. \] Let further \( \mathbb P_m \) be the set of all real algebraic polynomials of degree at most \(m\) and \[ \mathbb P_m (a_1, \ldots, a_n) := \left \{\frac{P(x)}{\sum_{k+1}^n |x - a_k|}, \;P \in \mathbb P_m \right \}. \] The main result of this article is Theorem 1.1. Let the nonreal elements in \(\{a_k\}\) be paired by complex conjugation. Then \(\{\mathbb P_{n-1} (a_1, \ldots, a_n)\}\) are dense in \(C[-1,1]\) if and only if \[ \sum \limits_{k=1}^{\infty} (1 - |c_k|) = \infty. \] This result extends the well-known theorem of Akhiezer.