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- Funk_transform abstract "In the mathematical field of integral geometry, the Funk transform (also called Minkowski–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, based on the work of Minkowski (1904). It is closely related to the Radon transform. The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.Classically, the Funk transform is defined as follows. Let ƒ be a continuous function on the 2-sphere in R3. Then, for a unit vector x, letwhere the integral is carried out with respect to the arclength ds of the great circle C(x) consisting of all unit vectors perpendicular to x:Clearly, the Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ƒ is even. In that case, the Funk transform takes even (continuous) functions to even continuous functions, and is furthermore invertible. As with the Radon transform, the inversion formula relies on the dual transform, defined byThis is the average value of the circle function ƒ over circles of arc distance p from the point x. The inverse transform is given byThe classical formulation is invariant under the rotation group SO(3). It is also possible to formulate the Funk transform in a manner that makes it invariant under the special linear group SL(3,R), due to (Bailey et al. 2003). Suppose that ƒ is a homogeneous function of degree −2 on R3. Then, for linearly independent vectors x and y, define a function φ by the line integraltaken over a simple closed curve encircling the origin once. The differential formis closed, which follows by the homogeneity of ƒ. By a change of variables, φ satisfiesand so gives a homogeneous function of degree −1 on the exterior square of R3,The function Fƒ : Λ2R3 → R agrees with the Funk transform when ƒ is the degree −2 homogeneous extension of a function on the sphere and the projective space associated to Λ2R3 is identified with the space of all circles on the sphere. Alternatively, Λ2R3 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}.".
- Funk_transform wikiPageExternalLink 980815.pdf.
- Funk_transform wikiPageID "23253927".
- Funk_transform wikiPageRevisionID "475078576".
- Funk_transform hasPhotoCollection Funk_transform.
- Funk_transform subject Category:Integral_geometry.
- Funk_transform subject Category:Integral_transforms.
- Funk_transform comment "In the mathematical field of integral geometry, the Funk transform (also called Minkowski–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, based on the work of Minkowski (1904). It is closely related to the Radon transform. The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.Classically, the Funk transform is defined as follows.".
- Funk_transform label "Funk transform".
- Funk_transform sameAs m.0660tl4.
- Funk_transform sameAs Q5509245.
- Funk_transform sameAs Q5509245.
- Funk_transform wasDerivedFrom Funk_transform?oldid=475078576.
- Funk_transform isPrimaryTopicOf Funk_transform.