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Bounded Point Derivations on Certain Function Spaces

Let 𝑋 be a compact subset of the complex plane and denote by 𝑅𝑝(𝑋) the closure of rational functions with poles off 𝑋 in the 𝐿𝑝(𝑋) norm. We show that if a point π‘₯0 admits a bounded point derivation on 𝑅𝑝(𝑋) for 𝑝 > 2, then there is an approximate derivative at π‘₯0. We also prove a similar result for higher order bounded point derivations. This extends a result of Wang, which was proven for 𝑅(𝑋), the uniform closure of rational functions with poles off 𝑋. In addition, we show that if a point π‘₯0 admits a bounded point derivation on 𝑅(𝑋) and if 𝑋 contains an interior cone, then the bounded point derivation can be represented by the difference quotient if the limit is taken over a non-tangential ray to π‘₯0. We also extend this result to the case of higher order bounded point derivations. These results were first shown by O'Farrell; however, we prove them constructively by explicitly using the Cauchy integral formula.

Identiferoai:union.ndltd.org:uky.edu/oai:uknowledge.uky.edu:math_etds-1055
Date01 January 2018
CreatorsDeterding, Stephen
PublisherUKnowledge
Source SetsUniversity of Kentucky
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceTheses and Dissertations--Mathematics

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