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Blaschke products and unwinding in higher dimensions

Ronald R. Coifman, Jacques Peyrière

Abstract

We give a necessary and sufficient condition for the convergence of an infinite product of rational inner functions on the polydisk, and explore generalization to the polydisk of Malmquist- Takenaka bases and various versions of unwinding

Blaschke products and unwinding in higher dimensions

Abstract

We give a necessary and sufficient condition for the convergence of an infinite product of rational inner functions on the polydisk, and explore generalization to the polydisk of Malmquist- Takenaka bases and various versions of unwinding
Paper Structure (15 sections, 4 theorems, 24 equations)

This paper contains 15 sections, 4 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Blaschke products
  4. Malmquist-Takenaka orthogonal systems
  5. Malmquist-Takenaka orthogonal systems
  6. Orthogonal projections
  7. Tensor products of Moebius functions
  8. Tensor product of two Möbius functions
  9. Tensor product of more Möbius functions
  10. Kernel for orthogonal projection
  11. An example
  12. Unwindings
  13. A first expansion
  14. Adaptative unwinding
  15. A less greedy unwinding

Key Result

Theorem 1

An infinite product $\prod_{n\ge 1} \beta(P_n) \frac{P_n^*}{P_n}$ converges (uniformly on compact subsets of ${\mathbb D}^d$) if and only if $\sum_{n\ge 1} \Bigl(1-\bigl|\alpha(P_n)\bigr|\Bigr)< +\infty.$

Theorems & Definitions (5)

  • Remark
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Corollary 1