Approximation by simple partial fractions with constraints on the poles. II (Q2817567)

This paper is a continuation of a recent article of the author [Sb. Math. 203, No. 11, 1553--1570 (2012); translation from Mat. Sb. 203, No. 11, 23--40 (2012; Zbl 1273.41012)]. Let \(K\) be a compact set with connected complement not separating the plane \(\mathbb{C}\), which lies in the union \(\widehat{E}\setminus E\) of the bounded components of the complement of another compact set \(E\). Denote by \(AC(K)\) the space of functions that are continuous on \(K\) and analytic in its interior. It is shown that the simple partial fractions with poles in \(E\) are dense in the space \(AC(K)\). Let \(E^+,E^-\subset\mathbb{C}\) be the connected components of the boundary of a doubly connected domain \(D \subset\overline{\mathbb{C}}\) and let \(K \subset\mathbb{ C}\setminus\overline{D}\) be a compact set with connected complement. Let \(\{r_1(z)\}\) be the simple partial fractions with poles in \(E^+\) and \(\{r_2(z)\}\) be the simple partial fractions with poles in \(E^-\). It is also shown that the differences \(r_1-r_2\) are dense in the space \(AC(K)\).