Funk Transform
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the mathematical field of integral geometry, the Funk transform is an integral transform defined by integrating a function on great circles of the sphere. It was introduced by Paul Funk in 1916. It is closely related to the Radon transform. The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere. The function F : 2R3 R agrees with the Funk transform when is the degree 2 homogeneous extension of a function on the sphere and...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the mathematical field of integral geometry, the Funk transform is an integral transform defined by integrating a function on great circles of the sphere. It was introduced by Paul Funk in 1916. It is closely related to the Radon transform. The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere. The function F : 2R3 R agrees with the Funk transform when is the degree 2 homogeneous extension of a function on the sphere and the projective space associated to 2R3 is identified with the space of all circles on the sphere. Alternatively, 2R3 can be identified with R3 in an SL(3,R)-invariant manner, and so the Funk transform F maps smooth even homogeneous functions of degree 2 on R3{0} to smooth even homogeneous functions of degree 1 on R3{0}.
