Bounded Point Derivations on Certain Function Spaces

Author

Stephen Deterding, University of KentuckyFollow

Author ORCID Identifier

https://orcid.org/0000-0003-2498-0245

Date Available

4-25-2018

Year of Publication

2018

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Mathematics

First Advisor

Dr. James Brennan

Abstract

Let 𝑋 be a compact subset of the complex plane and denote by 𝑅^𝑝(𝑋) the closure of rational functions with poles off 𝑋 in the 𝐿^𝑝(𝑋) norm. We show that if a point 𝑥₀ admits a bounded point derivation on 𝑅^𝑝(𝑋) for 𝑝 > 2, then there is an approximate derivative at 𝑥₀. 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 𝑥₀ 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 𝑥₀. 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