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Orthogonal polynomials in de Branges--Rovnyak spaces

Eugenio Dellepiane, Daniel Seco

Abstract

Given a function $b$, holomorphic on the disc and bounded by 1, one can construct an associated reproducing kernel Hilbert space called the de Branges--Rovnyak space $H(b)$. We explore representations of such spaces via descriptions of the corresponding families of orthogonal polynomials. We find relevant structures in the linear systems involved in a diversity of cases when $b$ is rational. We also establish a form of invariance under some composition operators on $H(b)$ spaces.

Orthogonal polynomials in de Branges--Rovnyak spaces

Abstract

Given a function , holomorphic on the disc and bounded by 1, one can construct an associated reproducing kernel Hilbert space called the de Branges--Rovnyak space . We explore representations of such spaces via descriptions of the corresponding families of orthogonal polynomials. We find relevant structures in the linear systems involved in a diversity of cases when is rational. We also establish a form of invariance under some composition operators on spaces.
Paper Structure (11 sections, 12 theorems, 103 equations)

This paper contains 11 sections, 12 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Main results
  4. A first family of examples
  5. Fricain--Mashreghi's work on Sarason's example
  6. Composition with a monomial
  7. Further remarks
  8. Recurrences and underlying matrix structure
  9. Poles of degree 1 in general position
  10. Higher order poles
  11. A potential description of the general phenomenon

Key Result

Theorem 1.2

Let $N\in\mathbb{N}\setminus\{0\}$ and $b(z)=(1+z^N)/2,z\in\mathbb{D}$. Then, the associated $H(b)$ space admits the following orthonormal basis of polynomials:

Theorems & Definitions (24)

  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.3
  • proof
  • ...and 14 more