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Inner functions and Pick-Nevanlinna interpolation in a multi-connected domain

Michel Crouzeix

TL;DR

The paper studies Pick–Nevanlinna interpolation on a multiply connected domain $\Omega$ by classifying inner functions of finite order and formulating optimization problems restricted to these inner functions. It proves the existence and uniqueness of a minimal-norm interpolant $f^*$ in $H^{\infty}(\Omega)$, with the normalization $h=f^*/m^*$ being an inner function of order at most $r{+}k{-}1$, and it extends the framework to Hermite interpolation and matrix interpolation. The results hinge on a constructive description of inner functions via harmonic measures and Green functions, and employ Fritz–John optimality conditions to derive order bounds and structural properties. The work connects to Abrahamse’s positive-definiteness criterion, providing a unified approach to generalized Pick–Nevanlinna problems on multiply connected domains and yielding a principled pathway for computing minimal-norm interpolants in this setting.

Abstract

We describe the set of inner functions of finite order in a multi-connected domain, then we consider an optimization formulation of the Pick-Nevanlinna interpolation problem, and we generalize it to Hermite type interpolation.

Inner functions and Pick-Nevanlinna interpolation in a multi-connected domain

TL;DR

The paper studies Pick–Nevanlinna interpolation on a multiply connected domain by classifying inner functions of finite order and formulating optimization problems restricted to these inner functions. It proves the existence and uniqueness of a minimal-norm interpolant in , with the normalization being an inner function of order at most , and it extends the framework to Hermite interpolation and matrix interpolation. The results hinge on a constructive description of inner functions via harmonic measures and Green functions, and employ Fritz–John optimality conditions to derive order bounds and structural properties. The work connects to Abrahamse’s positive-definiteness criterion, providing a unified approach to generalized Pick–Nevanlinna problems on multiply connected domains and yielding a principled pathway for computing minimal-norm interpolants in this setting.

Abstract

We describe the set of inner functions of finite order in a multi-connected domain, then we consider an optimization formulation of the Pick-Nevanlinna interpolation problem, and we generalize it to Hermite type interpolation.

Paper Structure

This paper contains 9 sections, 14 theorems, 95 equations.

Table of Contents

  1. 2000 Mathematical subject classifications :
  2. Introduction
  3. Inner functions of finite order
  4. A Schwarz type lemma
  5. An optimization problem with interpolation in $k$ points.
  6. Interpolation with minimal maximum norm
  7. Interpolation with derivatives
  8. Link with the original Pick-Nevanlinna problem
  9. Appendix. Proof of Lemma \ref{['lem7']}.

Key Result

Theorem 1

Let $A\in \mathbb C^{d,d}$ be a $d\times d$ matrix with eigenvalues in $\Omega$. Then, there exists an inner function $f_0$ of order $\leqslant d{+}k{-}1$ which realizes

Theorems & Definitions (26)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 16 more