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
Advisor
Dr. James Brennan
Abstract
Let ๐ be a compact subset of the complex plane and denote by ๐ ๐(๐) the closure of rational functions with poles off ๐ in the ๐ฟ๐(๐) norm. We show that if a point ๐ฅ0 admits a bounded point derivation on ๐ ๐(๐) for ๐ > 2, then there is an approximate derivative at ๐ฅ0. We also prove a similar result for higher order bounded point derivations. This extends a result of Wang, which was proven for ๐ (๐), the uniform closure of rational functions with poles off ๐. In addition, we show that if a point ๐ฅ0 admits a bounded point derivation on ๐ (๐) and if ๐ 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 ๐ฅ0. 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
