Document Type

Dissertation

Date of Degree

Spring 2013

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Palle Jorgensen

Second Advisor

Lihe Wang

Abstract

Let $\Omega\subset\mathbb{R}^n$ be an open and connected subset of $\mathbb{R}^n$. We say that a function $F\colon \Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, is positive definite if for any $x_1,\ldots,x_m\in\Omega$ and any $c_1,\ldots,c_m\in \mathbb{C}$ we have that $\sum_{j,k=1}^m F(x_j-x_k)c_j\overline{c_k}\geq 0$.

Let $F\colon\Omega-\Omega\to\mathbb{C}$ be a continuous positive definite function. We give necessary and sufficient conditions for $F$ to have an extension to a continuous and positive definite function defined on the entire Euclidean space $\mathbb{R}^n$. The conditions are formulated in terms of strong commutativity of some certain selfadjoint operators defined on a Hilbert space associated to our positive definite function.

Keywords

commuting selfadjoint operators, positive definite functions, reproducing kernel Hilbert spaces

Pages

iv, 80 pages

Bibliography

Includes bibliographical references (pages 78-80).

Copyright

Copyright 2013 Robert Niedzialomski

Included in

Mathematics Commons

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