The present survey paper is devoted to the spaces \(A^{-\infty }(\Omega )\) , their dual spaces, and their associated applications. Here, the domain \(\Omega \) is considered not only as the unit disk \(\mathbb {D}\) in the complex plane \(\mathbb {C}\) , but also as the broader class of the so-called Carathéodory domains in \(\mathbb {C}\) . Furthermore, this paper provides a detailed consideration of the representation problem, a topic that has remained largely untreated in the literature.
