A generalization of the Funk–Radon transform to circles passing through a fixed point
The Funk–Radon transform assigns to a function on the two-sphere its mean values along all great circles. We consider the following generalization: we replace the great circles by the small circles being the intersection of the sphere with planes containing a common point ζ inside the sphere. If ζ i...
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| Format: | Others |
| Language: | English |
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Universitätsbibliothek Chemnitz
2016
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| Online Access: | http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-192513 http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-192513 http://www.qucosa.de/fileadmin/data/qucosa/documents/19251/Preprint_2015_17.pdf http://www.qucosa.de/fileadmin/data/qucosa/documents/19251/signatur.txt.asc |
| Summary: | The Funk–Radon transform assigns to a function on the two-sphere its mean values along all great circles. We consider the following generalization: we replace the great circles by the small circles being the intersection of the sphere with planes containing a common point ζ inside the sphere. If ζ is the origin, this is just the classical Funk–Radon transform. We find two mappings from the sphere to itself that enable us to represent the generalized Radon transform in terms of the Funk–Radon transform. This representation is utilized to characterize the nullspace and range as well as to prove an inversion formula of the generalized Radon transform. |
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