Harmonic Analysis of Radon Filtrations for Sn and Its q-Analogue GLn(q)

We present a unified approach to the study of Radon transforms related to the group Sn and its q-analogue GLn(q). In both cases, in a uniform way, we define a sequence of generalized Radon transforms that are intertwining operators for natural representations associated to Gel'fand spaces for our groups. This sequence and the sequence of their adjoint enable us to decompose in a recursive way these natural representations into irreducibles and to compute explicitly the associated spherical functions. Our methods and results rely then strongly on q-analogy.

Multilinear generalized Radon transforms and point configurations

We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving (

A fast butterfly algorithm for generalized Radon transforms

Generalized Radon transforms, such as the hyperbolic Radon transform, cannot be implemented as efficiently in the frequency domain as convolutions, thus limiting their use in seismic data processing. We have devised a fast butterfly algorithm for the hyperbolic Radon transform. The basic idea is to reformulate the transform as an oscillatory integral operator and to construct a blockwise low-rank approximation of the kernel function. The overall structure follows the Fourier integral operator butterfly algorithm. For 2D data, the algorithm runs in complexity [Formula: see text], where [Formula: see text] depends on the maximum frequency and offset in the data set and the range of parameters (intercept time and slowness) in the model space. From a series of studies, we found that this algorithm can be significantly more efficient than the conventional time-domain integration.