Analysis and Mathematical Physics

, Volume 3, Issue 3, pp 201–234

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

Article

DOI: 10.1007/s13324-013-0054-9

Cite this article as:
Brennan, J.E. & Mattingly, C.N. Anal.Math.Phys. (2013) 3: 201. doi:10.1007/s13324-013-0054-9

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 \(p,\, 1\le p<\infty \), let \(R^p(X)\) be the closure of \(R(X)\) in the \(L^p(X, dA)\)-norm. Since \(X\) has no interior \(R^p(X)=L^p(X)\) whenever \(1\le p <2\), 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 \(p, \, 2\le p < \infty \).

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

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