Complex horospherical transform on real sphere
Simon Gindikin · Contemporary mathematics - American Mathematical Society · 2005
Abstract. We define a new integral transform on the real sphere which is invariant relative to the orthogonal group and similar to the horospherical Radon transform for the hyperbolic space. This transform involves complex geometry associated with the sphere. In integral geometry on hyperbolic spaces and other noncompact symmetric spaces there are 2 versions of the Radon transform: geodesic and horospheric [GGG03]. The horospherical Radon transform has more essential connections with the harmonic analysis on these spaces than its geodesic analogue. The geodesic version of the Radon transform is well known on the sphere. It is the famous Minkowsky-Funk transform of integration along the big subspheres [GGG03]. Let us recall that this transform was discovered earlier than the Radon transform. We want in this note to construct the analogue of the horospherical transform on the sphere. At first glance it looks strange since there are no horospheres on the sphere, but we will show that such a transform exists if in the geometrical background we replace real horospheres by complex ones. We will follow the idea of [Gi00] which