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RATIONAL APPROXIMATION ON COMPACT NOWHERE DENSE SETSMattingly, Christopher 01 January 2012 (has links)
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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<i>L<sup>p</sup></i> Bounded Point Evaluations for Polynomials and Uniform Rational ApproximationMilitzer, Erin 01 January 2010 (has links)
A connection is established between uniform rational approximation, and approximation in the mean by polynomials on compact nowhere dense subsets of the complex plane C. Peak points for R(X) and bounded point evaluations for Hp(X, dA), 1 ≤ p < ∞, play a fundamental role.
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