Let π be a compact subset of the complex plane and denote by π π(π) the closure of rational functions with poles off π in the πΏπ(π) norm. We show that if a point π₯0 admits a bounded point derivation on π π(π) for π > 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 π (π), the uniform closure of rational functions with poles off π. In addition, we show that if a point π₯0 admits a bounded point derivation on π (π) and if π 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.
| Identifer | oai:union.ndltd.org:uky.edu/oai:uknowledge.uky.edu:math_etds-1055 |
| Date | 01 January 2018 |
| Creators | Deterding, Stephen |
| Publisher | UKnowledge |
| Source Sets | University of Kentucky |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Theses and Dissertations--Mathematics |
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