Approximation by rational functions on compact nowhere dense subsets of the complex plane

J. E. Brennan, C. N. Mattingly

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Let X be a compact nowhere dense subset of the complex plane, and let dA denote two-dimensional or area measure on X. Let R(X) denote the uniform closure of the rational functions having no poles on X, and for each (Formula presented.), let Rp(X) be the closure of R(X) in the Lp(X, dA)-norm. Since X has no interior Rp(X)=Lp(X) whenever (Formula presented.), but for p=2 a kind of phase transition occurs that can be quite striking at times. Our main goal here is to study the manner in which similar phase transitions can occur at any value of (Formula presented.).

Original languageEnglish
Pages (from-to)201-234
Number of pages34
JournalAnalysis and Mathematical Physics
Volume3
Issue number3
DOIs
StatePublished - Sep 25 2013

Bibliographical note

Publisher Copyright:
© 2013, Springer Basel.

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Mathematical Physics

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