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Alexander, J. Ralph Rogers, Allen Dale 2011-05-07T13:24:15Z 2011-05-07T13:24:15Z 10000-01-01 1990 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. Made available in DSpace on 2011-05-07T13:24:15Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114391.pdf: 2556930 bytes, checksum: 2d5370ab326be59570ae809cf2acd27d (MD5) Previous issue date: 1990 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:54:19Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:25:13-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9114391 (UMI)AAI9114391 http://hdl.handle.net/2142/21952 eng Copyright 1990 Rogers, Allen Dale Mathematics Theory and applications of a functional from metric geometry text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.