Date of Award

2024

Degree Name

Mathematics

College

College of Science

Type of Degree

M.A.

Document Type

Thesis

First Advisor

Dr. Stephen Deterding, Committee Chairperson

Second Advisor

Dr. Tom Cuchta

Third Advisor

Dr. Anna Mummert

Abstract

This paper is concerned primarily with a type of subset of the complex plane known as a Roadrunner set, and its admittance of a bounded point derivation with respect to a given norm on the complex plane. The four norms we are concerned with are the Uniform norm, Lipschitz norm, Lp norm, and Campanato semi-norm. The purpose of this thesis is to provide researchers in approximation theory with more tools for them to accomplish their goals such as the proof of theorems regarding generalized derivatives. A connection has historically been established between the existence of bounded point derivations and the convergence of an infinite series by means of the notion of a capacity. What follows is a series of implications for first-order bounded point derivations with respect to the four aforementioned norms. The admittance of a tth-order bounded point derivation with respect to one norm does not relate as clearly to other norms, but a relation can be formed nonetheless. Sets that admit a bounded point derivation of arbitrary order are considered as well.

Subject(s)

Function spaces.

Functional analysis.

Mathematics.

Approximation theory.

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