Asymptotically fast algorithms for spherical and related transforms

The problem of computing the convolution of two functions on the sphere by means of a spherical transform is considered. Such convolutions are applicable to surface recognition and the location of both rotated and translated patterns in an image. The authors give convolution theorems that relate the spherical transform to convolution, sampling theorems that allow the exact computation of the transform for band-limited functions, and algorithms with asymptotically improved running time for the exact computation of the harmonic expansion. The net result is an O(n^1.5(log n)²) algorithm for the exact computation of the convolution of two bandlimited functions sampled at n points in accordance with the sampling theorem. The techniques developed are applicable to computing other transforms, such as the Laguerre, Hermite, and Hankel transforms