Continuous Regularized Least Squares Polynomial Approximation on the Sphere

In this paper, we consider the problem of polynomial reconstruction of smooth functions on the sphere from their noisy values at discrete nodes on the two-sphere. The method considered in this paper is a weighted least squares form with a continuous regularization. Preliminary error bounds in terms of regularization parameter, noise scale, and smoothness are proposed under two assumptions: the mesh norm of the data point set and the perturbation bound of the weight. Condition numbers of the linear systems derived by the problem are discussed. We also show that spherical - designs, which can be seen as a generalization of spherical - designs, are well applied to this model. Numerical results show that the method has good performance in view of both the computation time and the approximation quality.