In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integrating a function on great circles of the sphere. It was introduced by Paul Funk in 1911, 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.
Definition The Funk transform is defined as follows. Let ƒ be a continuous function on the d-1-sphere Sd-1 in Rd. Then, for a unit vector x, let
F f ( x ) = ∫ u ∈ C ( x ) f ( u ) d s ( u ) {\displaystyle 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 ( x ) = { u ∈ S d − 1 ∣ u ⋅ x = 0 } . {\displaystyle C(\mathbf {x} )=\{\mathbf {u} \in S^{d-1}\mid \mathbf {u} \cdot \mathbf {x} =0\}.}
Inversion 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.
Spherical harmonics Every square-integrable function f ∈ L 2 ( S 2 ) {\displaystyle f\in L^{2}(S^{2})} on the sphere can be decomposed into spherical harmonics Y n k {\displaystyle Y_{n}^{k}}
f = ∑ n = 0 ∞ ∑ k = − n n f ^ ( n , k ) Y n k . {\displaystyle f=\sum _{n=0}^{\infty }\sum _{k=-n}^{n}{\hat {f}}(n,k)Y_{n}^{k}.}
Then the Funk transform of f reads
F f = ∑ n = 0 ∞ ∑ k = − n n P n ( 0 ) f ^ ( n , k ) Y n k {\displaystyle Ff=\sum _{n=0}^{\infty }\sum _{k=-n}^{n}P_{n}(0){\hat {f}}(n,k)Y_{n}^{k}}
where P 2 n + 1 ( 0 ) = 0 {\displaystyle P_{2n+1}(0)=0} for odd values and
P 2 n ( 0 ) = ( − 1 ) n 1 ⋅ 3 ⋅ 5 ⋯ 2 n − 1 2 ⋅ 4 ⋅ 6 ⋯ 2 n = ( − 1 ) n ( 2 n − 1 ) ! ! ( 2 n ) ! ! {\displaystyle P_{2n}(0)=(-1)^{n}\,{\frac {1\cdot 3\cdot 5\cdots 2n-1}{2\cdot 4\cdot 6\cdots 2n}}=(-1)^{n}\,{\frac {(2n-1)!!}{(2n)!!}}}
for even values. This result was shown by Funk (1913).
Helgason's inversion formula Another inversion formula is due to Helgason (1999). As with the Radon transform, the inversion formula relies on the dual transform F* defined by
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