RATIONAL APPROXIMATION ON COMPACT NOWHERE DENSE SETS

RATIONAL APPROXIMATION ON COMPACT NOWHERE DENSE SETS

For a compact, nowhere dense set X in the complex plane, C, define Rp(X) as the closure of the rational functions with poles off X in Lp(X, dA). It is well known that for 1 ≤ p < 2, Rp(X) = Lp(X) . Although density may not be achieved for p > 2, there exists a set X so that Rp(X) = Lp(X) for p...

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Bibliographic Details
Main Author: Mattingly, Christopher
Format: Others
Published: UKnowledge 2012
Subjects:
uniform and Lp rational approximation
q > capacity
bounded point evaluations
representing measures
Applied Mathematics
Mathematics
Online Access:http://uknowledge.uky.edu/math_etds/4
http://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1003&amp;context=math_etds
Description
Summary:For a compact, nowhere dense set X in the complex plane, C, define Rp(X) as the closure of the rational functions with poles off X in Lp(X, dA). It is well known that for 1 ≤ p < 2, Rp(X) = Lp(X) . Although density may not be achieved for p > 2, there exists a set X so that Rp(X) = Lp(X) for p up to a given number greater than 2 but not after. Additionally, when p > 2 we shall establish that the support of the annihiliating and representing measures for Rp(X) lies almost everywhere on the set of bounded point evaluations of X.

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