TY - JOUR
T1 - Approximation by rational functions on compact nowhere dense subsets of the complex plane
AU - Brennan, J. E.
AU - Mattingly, C. N.
PY - 2013/9/25
Y1 - 2013/9/25
N2 - 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.).
AB - 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.).
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U2 - 10.1007/s13324-013-0054-9
DO - 10.1007/s13324-013-0054-9
M3 - Article
AN - SCOPUS:84937923462
SN - 1664-2368
VL - 3
SP - 201
EP - 234
JO - Analysis and Mathematical Physics
JF - Analysis and Mathematical Physics
IS - 3
ER -