# Theory and applications of a functional from metric geometry

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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