Distribution of Points on the Sphere and Spherical Designs.

摘要

This thesis concentrates on distribution of points on the unit sphere and polynomial approximation on the unit sphere by using spherical designs.;A set of N points on the unit sphere is a spherical t-design if the average value of any polynomial of degree at most t over the XN is equal to the average value of the polynomial over the sphere.;The main contributions of this thesis consist of the following two parts. 1. We consider the characterization and computation of spherical t-designs on the unit sphere in 3 dimensional space when N ≥ (t + 1)², the dimension of the space of spherical polynomials of degree at most t. We show how construct well conditioned spherical t-designs with N ≥ (t + 1)² points by maximizing the determinant of a Gram matrix which satisfies undetermined nonlinear equations. Interval methods are then used to prove the existence of a true spherical t-design and to provide a guaranteed interval containing the true spherical t-design. The resulting spherical designs have good geometrical properties (separation and mesh norm). We discuss desirable properties of the points for both equal weight numerical integration and polynomial interpolation on the sphere, and give examples to illustrate the characterization of these points. 2, We consider polynomial approximation on the unit sphere by a class of regularized discrete least squares methods, with novel choices for the regularization operators and the point sets of the discretization. We allow different kinds of rotationally invariant regularization operators, including the zero operator (in which case the approximation includes interpolation, quasi-interpolation and hyperinterpolation); powers of the negative Laplace-Beltrami operator (which can be suitable when there are data errors); and regularization operators that yield filtered polynomial approximations. As node sets we use spherical t-designs. For t ≥ 2L and an approximation polynomial of degree L it turns out that there is no linear algebra problem to be solved. Finally, we give numerical examples to illustrate the theoretical results and show that well chosen regularization operators and well conditioned spherical t-designs can provide good polynomial approximation on the sphere, with exact data or contaminated data.