This thesis can be broadly categorized under the theories of multi-variable operators and holomorphic functions.

The first part investigates the relationship between two classical theorems: the Carathéodory approximation theorem and the Herglotz representation theorem, showing their equivalence in classical settings. We then extend the Carathéodory approximation to finitely connected domains for operator-valued functions. Additionally, using the Carathéodory approximation, an integral representation for Herglotz functions in finitely connected planar domains is derived in the style of Korányi and Pukánszky. Furthermore, we analyze operator-valued Herglotz functions using Krein space theory and Kolmogorov decomposition, obtaining realization formulae in a general setting encompassing various domains such as the polydisc, the annulus, certain quotient domains, and certain distinguished varieties in the bidisc, like the Neil parabola.

The second part begins by developing the function theory on quotients of the polydisc. Studying the algebra generated by inner functions, Carathéodory approximation, Fisher type approximation, and the structure of rational inner functions are some of the main highlights. Then we focus on Hankel operators on Hardy spaces associated with quotient domains obtained by the action of finite pseudo-reflection groups on bounded symmetric domains. We characterize the boundedness and compactness of small Hankel operators and address the failure of Nehari’s theorem for big Hankel operators on the Hardy space of the polydisc, a result previously proven by Cotlar and Sadosky, Bakonyi and Timotin, and Ahern and Youssfi. This phenomenon extends to quotients of the polydisc and the Euclidean ball, leading to algebraic consequences in the category of Hilbert modules over the uniform algebra of holomorphic functions in the interior of quotient domains and continuous on the closure. The thesis shows that Hardy modules in this setting are not projective but identifies a restricted category (imposing some topological conditions) where projective objects exist. The Carathéodory approximation for quotient domains plays a crucial role in proving projectivity in this smaller category.