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Author ORCID Identifier
Date Available
4-25-2018
Year of Publication
2018
Document Type
Doctoral Dissertation
Degree Name
Doctor of Philosophy (PhD)
College
Arts and Sciences
Department/School/Program
Mathematics
Faculty
Dr. James Brennan
Faculty
Dr. Peter Hislop
Abstract
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.
Digital Object Identifier (DOI)
https://doi.org/10.13023/ETD.2018.143
Recommended Citation
Deterding, Stephen, "Bounded Point Derivations on Certain Function Spaces" (2018). Theses and Dissertations--Mathematics. 51.
https://uknowledge.uky.edu/math_etds/51
