Abstract

The Herglotz representation and the Caratheodory approximation are classical theorems in complex analytic function theory. The first is about the integral representation of a holomorphic function on the open unit disc having non-negative real part, and the second is about approximating a bounded holomorphic function with sup-norm less than or equal to one on the open unit disc by rational inner functions.

In this talk, we shall discuss the equivalence of these two theorems on the disc and the finitely connected planar domains. We shall see a proof of matrix-valued Caratheodory approximation on the open unit disc using the Herglotz representation. 

Also, we give a different Herglotz representation on the finitely connected planar domain using the Szego kernel of the Hardy space of such domain.