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Herglotz's representation and Caratheodory's approximation

Tirthankar Bhattacharyya, Mainak Bhowmik, Poornendu Kumar

Abstract

Herglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well-known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi-connected domain $Ω$, a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Koranyi and Pukanszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for matrix-valued functions on a multi-connected domain.

Herglotz's representation and Caratheodory's approximation

Abstract

Herglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well-known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi-connected domain , a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Koranyi and Pukanszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for matrix-valued functions on a multi-connected domain.
Paper Structure (14 sections, 11 theorems, 93 equations)

This paper contains 14 sections, 11 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Carathéodory's theorem for operator-valued functions on the disc
  3. Herglotz's representation implies Carathéodory's approximation
  4. Carathéodory's approximation implies Herglotz's representation
  5. Carathéodory and Herglotz in multi-connected domains
  6. Carathéodory's theorem in multi-connected domains
  7. Properties of the Szegö Kernel
  8. The Koranyi-Pukanszky formula
  9. The equivalence
  10. Carathéodory's theorem for matrix-valued functions
  11. Integral representation for operator-valued Herglotz functions
  12. Uniqueness of Herglotz's representation in scalar case:
  13. Extension of Herglotz's representation to operator-valued case:
  14. An exceptional domain

Key Result

Theorem 1.2

Any holomorphic function $\varphi: \mathbb{D} \to \overline{\mathbb{D}}$ can be approximated (uniformly on compacta in $\mathbb{D}$) by rational inner functions, i.e., rational functions whose poles are outside $\overline{\mathbb D}$ and which assume unimodular values on the unit circle.

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.2: Carathéodory
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • ...and 11 more