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Title:Theory and applications of a functional from metric geometry
Author(s):Rogers, Allen Dale
Doctoral Committee Chair(s):Alexander, J. Ralph
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
Abstract:For positive $\alpha$, and for complex measures $\mu$ and $\nu$ on R$\sp{n}$, define $J\sp\alpha(\mu,\nu)=\int\int\vert x-y\vert\sp\alpha\ d\mu(x)d\bar \nu(y)$. Study of the energy integral $J\sp\alpha$ has its roots in metric embedding theory and potential theory. Subject to certain moment vanishing conditions, a representation formula is proved using the Fourier transform and tempered distributions. Ideas from integral geometry are used to apply the functional $J\sp\alpha$ to irregularities of distribution, and estimates of discrepancy are obtained for measures that do not have atoms. Finally, the close relation between $J\sp1$ and the Radon transform is investigated. Inequalities are deduced that bound certain norms of the Radon transform away from zero.
Issue Date:1990
Type:Text
Language:English
URI:http://hdl.handle.net/2142/21952
Rights Information:Copyright 1990 Rogers, Allen Dale
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9114391
OCLC Identifier:(UMI)AAI9114391


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