Spherical deconvolution

This paper proposes nonparametric deconvolution density estimation over S-2. Here we would think of the S-2 elements of interest being corrupted by random SO(3) elements (rotations). The resulting density on the observations would be a convolution of the SO(3) density with the true S-2 density. Consequently, the methodology, as in the Euclidean case, would be to use Fourier analysis on SO(3) and S-2, involving rotational and spherical harmonics, respectively. We especially consider the case where the deconvolution operator is a bounded operator lowering the Sobolev order by a finite amount. Consistency results are obtained with rates of convergence calculated under the expected L-2 and Sobolev square norms that are proportionally inverse to some power of the sample size. As an example we introduce the rotational version of the Laplace distribution. (C) 1998 Academic Press