Removability of exceptional sets for differentiable and Lipschitz functions

Article, Preprint, Contribution for newspaper or weekly magazine English OPEN
Craig, J. ; Feinstein, J. F. ; Patrick, P. (2014)
  • Publisher: American Mathematical Society
  • Related identifiers: doi: 10.1090/conm/645
  • Subject: Primary 26A27, 26A16, 26A24, Secondary 46J15, 46J10, 46J05 | 46J10 (Primary), 54H99 (Secondary) | Spectral isometries, Jordan isomorphisms, commutative Banach algebras | Mathematics - Functional Analysis
    arxiv: Statistics::Computation | Mathematics::Algebraic Topology | Physics::Accelerator Physics | Quantitative Biology::Genomics | Statistics::Methodology

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.
  • References (7)

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    [7] X. Tolsa, Painlev´e's problem, analytic capacity and curvature of measures, European Congress of Mathematics, 459-476, Eur. Math. Soc., Zu¨rich, 2005. School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, UK Current address: KPMG LLP, 8 Salisbury Square, London EC4Y 8BB, UK E-mail address: James.Craig@kpmg.co.uk School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, UK E-mail address: Joel.Feinstein@nottingham.ac.uk School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, UK E-mail address: pt.patrick@outlook.com

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