球面混合插值的逼近性质

球面径向基函数(SBF)和多项式样条函数均为处理球面散乱数据的有效工具. 本文考虑由球面径向基函数与球面多项式函数组成的混合插值模型, 并利用最小二乘法求解该模型. 对于该插值模型, 首先, 给出带Bessel势的Sobolev空间中的Bernstein不等式, 然后利用该不等式建立逼近正定理,并进一步给出该插值工具的误差估计. 最后, 研究该插值方式(即利用最小二乘法求解混合插值模型)的稳定性.

Approximation properties of hybrid interpolation on spheres

The interpolation of spherical radial basis functions (SBF) and the polynomial splines are both efficient methods for modeling of functions based on scattered data on the unit sphere. In this paper, we are interested in employing hybrid method for scattered data modeling, and solving the system with linear leastsquares method. Making use of Bernstein inequalities, we establish a direct theorem in Bessel-potential Sobolev spaces. Furthermore, we give the interpolation error estimate. Finally, we compare its stability with interpolation by radial basis functions.