Radon transform on affine buildings of rank three

Abstract

We define the Radon transform for functions on the set of chambers of affine, locally finite, rank three buildings. We investigate the problem of the inversion of this transform. Explicit inversion formulas are exhibited for functions which fulfill required summability conditions.

Reflections acting efficiently on a building

“…It also extends some of the ideas found in [10] and [11]. See [1,2] for different examples of Radon transforms associated with buildings.…”

Radon transforms on Ãn-buildings

“…We study the problem of the inversion of R. In Section 1 we show that for 0 < k < n the operator R is not injective. Then, to complete the work done in [3], in Section 2 we are concerned about the problem of inverting R when n = 2 and k = 0. We show how this problem is related to the problem of inverting the Laplacian.…”

Radon Transforms in Hyperbolic Spaces and Their Discrete Counterparts

“…In particular, the horospherical Radon transform on trees has been studied in [7,12,15,20] and the X-ray transform in [1,6]. Homogeneous trees do not provide a discrete model for n-dimensional hyperbolic space except when n = 2 (that is, for the disc); discrete models for higher dimensional hyperbolic spaces should be obtained from higher-rank Bruhat-Tits buildings, see [3,4].…”