DOI

10.17077/etd.a5iwlm7p

Document Type

Dissertation

Date of Degree

Spring 2013

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Jorgensen, Palle

Second Advisor

Wang, Lihe

First Committee Member

Jorgensen, Palle

Second Committee Member

Wang, Lihe

Third Committee Member

Frohman, Charles

Fourth Committee Member

Muhly, Paul

Fifth Committee Member

Strohmer, Gerhard

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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