The Funk transform as a Penrose transform
Date
1999
Authors
Bailey, Toby N.
Eastwood, Michael George
Gover, A. Rod
Mason, Lionel J.
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Journal article
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Mathematical Proceedings of the Cambridge Philosophical Society, 1999; 125(1):67-81
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By Toby N. Bailey Michael G. Eastwood, A. Rod Gover, and Lionel J. Mason
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Abstract
The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S²[subset or is implied by][open face R]³ to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [open face C][open face P]₂ to [open face C][open face P]*ast;₂. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.
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© 1999 Cambridge Philosophical Society