Theory and applications of a functional from metric geometry
Rogers, Allen Dale
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https://hdl.handle.net/2142/21952
Description
Title
Theory and applications of a functional from metric geometry
Author(s)
Rogers, Allen Dale
Issue Date
1990
Doctoral Committee Chair(s)
Alexander, J. Ralph
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
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