Cubature formulas and discrete fourier transform on compact manifolds

Analysis on two dimensional surfaces and in particular on the sphere S found many applications in computerized tomography, statistics, signal analysis, seismology, weather prediction, and computer vision. During last years many problems of classical harmonic analysis were developed for functions on manifolds and especially for functions on spheres: splines, interpolation, approximation, different aspects of Fourier analysis, continuous and discrete wavelet transform, quadrature formulas. Our list of references is very far from being complete [1]-[6], [8]-[16], [19]-[35]. More references can be found in monographs [12], [20]. The goal of the paper is to describe three types of cubature formulas on general compact Riemannian manifolds which require essentially optimal number of nodes. Cubature formulas introduced in section 3 are exact on subspaces of band-limited functions. Cubature formulas constructed in section 4 are exact on spaces of variational splines and, at the same time, asymptotically exact on spaces of band-limited functions. In section 5 we prove existence of cubature formulas with positive weights which are exact on spaces of band-limited functions. In section 7 we prove that on homogeneous compact manifolds the product of two band-limited functions is also band-limited. This result makes our findings about