Funk transform

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.

The Funk transform is defined as follows. Let ƒ be a continuous function on the 2-sphere S² in R³. Then, for a unit vector x, let

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:

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.

Every square-integrable function ${\displaystyle f\in L^{2}(S^{2})}$ on the sphere can be decomposed into spherical harmonics ${\displaystyle Y_{n}^{k}}$

...
Wikipedia