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Simultaneous Approximation by Finite Blaschke Products and Bounded Universal Functions

Konstantinos Maronikolakis

Abstract

This paper complements the work done on simultaneous approximation results in classical Banach spaces, by focusing on approximation by finite Blaschke products. We prove the existence of a finite Blaschke product that approximates a prescribed holomorphic function bounded by 1 locally uniformly on the unit disc, and simultaneously approximates a prescribed unimodular continuous function uniformly on a compact subset of the unit circle of arclength measure 0. We also prove an analogue where the continuous function is bounded by 1 and the the approximation is achieved by an appropriate dilate of the finite Blaschke product. These results are essentially combinations of classical results of Caratheodory and Fisher on approximation by finite Blaschke products. We also give analogues for singular inner functions. Finally, we apply our results to prove the existence of bounded holomorphic functions on the unit disc that exhibit a certain universal boundary behaviour.

Simultaneous Approximation by Finite Blaschke Products and Bounded Universal Functions

Abstract

This paper complements the work done on simultaneous approximation results in classical Banach spaces, by focusing on approximation by finite Blaschke products. We prove the existence of a finite Blaschke product that approximates a prescribed holomorphic function bounded by 1 locally uniformly on the unit disc, and simultaneously approximates a prescribed unimodular continuous function uniformly on a compact subset of the unit circle of arclength measure 0. We also prove an analogue where the continuous function is bounded by 1 and the the approximation is achieved by an appropriate dilate of the finite Blaschke product. These results are essentially combinations of classical results of Caratheodory and Fisher on approximation by finite Blaschke products. We also give analogues for singular inner functions. Finally, we apply our results to prove the existence of bounded holomorphic functions on the unit disc that exhibit a certain universal boundary behaviour.

Paper Structure

This paper contains 7 sections, 9 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Approximation by finite Blaschke products
  3. The theorems of Fisher and Caratheodory and main results
  4. Proof of Theorem \ref{['sim_approx_circle']}
  5. Proof of Theorem \ref{['sim_approx_disc']}
  6. Approximation by singular inner functions
  7. Universal radial approximation

Key Result

Theorem 2.1

Let $K$ be a compact subset of ${\mathbb{T}}$ of arclength measure $0$ and $L$ be a compact subset of ${\mathbb{D}}$. Let also $\varphi$ be a continuous function on $K$ with $|\varphi|\equiv1$ on $K$ and $f\in\mathcal{B}$. Finally, let $\varepsilon>0$. Then, there exists a finite Blaschke product $B

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • proof : Proof of Theorem \ref{['sim_approx_circle']}
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 12 more