Funk transform

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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, let

$Ff({\mathbf {x}})=\int _{{{\mathbf {u}}\in C({\mathbf {x}})}}f({\mathbf {u}})\,ds({\mathbf {u}})$

where 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:

$C({\mathbf {x}})=\{{\mathbf {u}}\in S^{2}\mid {\mathbf {u}}\cdot {\mathbf {x}}=0\}.$

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 by

$(F^{*}f)(p,{\mathbf {x}})={\frac {1}{2\pi \cos p}}\int _{{\|{\mathbf {u}}\|=1,{\mathbf {x}}\cdot {\mathbf {u}}=\sin p}}f({\mathbf {u}})\,|d{\mathbf {u}}|.$

This is the average value of the circle function ƒ over circles of arc distance p from the point x. The inverse transform is given by

$f({\mathbf {x}})={\frac {1}{2\pi }}\left\{{\frac {d}{du}}\int _{0}^{u}F^{*}(Ff)(\cos ^{{-1}}v,{\mathbf {x}})v(u^{2}-v^{2})^{{-1/2}}\,dv\right\}_{{u=1}}.$

The 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 integral

$\varphi ({\mathbf {x}},{\mathbf {y}})={\frac {1}{2\pi }}\oint f(u{\mathbf {x}}+v{\mathbf {y}})(u\,dv-v\,du)$

taken over a simple closed curve encircling the origin once. The differential form

$f(u{\mathbf {x}}+v{\mathbf {y}})(u\,dv-v\,du)$

is closed, which follows by the homogeneity of ƒ. By a change of variables, φ satisfies

$\phi (a{\mathbf {x}}+b{\mathbf {y}},c{\mathbf {x}}+d{\mathbf {y}})={\frac {1}{|ad-bc|}}\phi ({\mathbf {x}},{\mathbf {y}}),$

and so gives a homogeneous function of degree −1 on the exterior square of R³,

$Ff({\mathbf {x}}\wedge {\mathbf {y}})=\phi ({\mathbf {x}},{\mathbf {y}}).$

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}.

References

Bailey, T. N.; Eastwood, Michael G.; Gover, A. Rod; Mason, L. J. (2003), "Complex analysis and the Funk transform", Journal of the Korean Mathematical Society 40 (4): 577–593, doi:10.4134/JKMS.2003.40.4.577, MR 1995065
Dann, Susanna (2010), On the Minkowski-Funk Transform, arXiv:1003.5565
Funk, Paul (1915), "Über eine geometrische Anwendung der Abelschen Integralgleichung", Mathematische Annalen 77 (1): 129–135, doi:10.1007/BF01456824, MR 1511851 .
Guillemin, Victor (1976), "The Radon transform on Zoll surfaces", Advances in Mathematics 22 (1): 85–119, doi:10.1016/0001-8708(76)90139-0, MR 0426063 .
Helgason, Sigurdur (1999), The Radon transform, Progress in Mathematics 5 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4109-2, MR 1723736 .
Minkowski, Hermann (1904), "About bodies of constant width", Mathematics Sbornik 25: 505–508

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Funk transform

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References