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Swiss Cheeses and Their Applications

J. F. Feinstein, S. Morley, H. Yang

Abstract

Swiss cheese sets have been used in the literature as useful examples in the study of rational approximation and uniform algebras. In this paper, we give a survey of Swiss cheese constructions and related results. We describe some notable examples of Swiss cheese sets in the literature. We explain the various abstract notions of Swiss cheeses, and how they can be manipulated to obtain desirable properties. In particular, we discuss the Feinstein-Heath classicalisation theorem and related results. We conclude with the construction of a new counterexample to a conjecture of S. E. Morris, using a classical Swiss cheese set.

Swiss Cheeses and Their Applications

Abstract

Swiss cheese sets have been used in the literature as useful examples in the study of rational approximation and uniform algebras. In this paper, we give a survey of Swiss cheese constructions and related results. We describe some notable examples of Swiss cheese sets in the literature. We explain the various abstract notions of Swiss cheeses, and how they can be manipulated to obtain desirable properties. In particular, we discuss the Feinstein-Heath classicalisation theorem and related results. We conclude with the construction of a new counterexample to a conjecture of S. E. Morris, using a classical Swiss cheese set.

Paper Structure

This paper contains 12 sections, 17 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Swiss cheeses
  4. Examples of Swiss cheese sets
  5. Bounded point derivations
  6. Regular uniform algebras
  7. Other examples
  8. Classicalisation theorems
  9. Comparison of Swiss cheeses
  10. Semiclassicalisation
  11. A classical counterexample to the conjecture of S. E. Morris
  12. Open questions

Key Result

Proposition 2.6

Let $X$ and $Y$ be compact plane sets with $Y\subseteq X$. Then$:$

Figures (4)

  • Figure 1: A classical Swiss cheese set where the discrepancy can be improved.
  • Figure 2: Elementary lemmas for combining and pulling in disks.
  • Figure 3: Length of overlap and extrusion length shown by the unbroken line.
  • Figure 4: A pair $(K_n,U_n)$ as in the proof of Theorem \ref{['SEMorris']}.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • ...and 17 more