光顺样条函数的再生核表示与计算

光顺样条是散乱数据拟合的理想函数,是噪声数据最优平滑的重要工具。因此,光顺样条的数学表示和计算的研究具有重要的意义。本文在一般的线性微分算子和线性泛函的情况下讨论光顺样条函数的构造和计算,通过构造一个适当的再生核Hilbert空间,使得所讨论的微分算子光顺样条成为该空间中的最小范数问题,再利用投影理论建立了光顺样条函数的再生核表示方法,并得到了插值偏差表达式。作为特例,还给出了奇次多项式光顺样条函数新的简洁的计算方法。

Construction and Computation of Smoothing Splines Via Reproducing Kernels

Smoothing splines are well known to be the ideal functions for fitting of discrete data, and also the effective method for smoothing noisy data. Therefore, it is very important to study the construction and computation of smoothing splines. In this paper, the construction and computation of smoothing splines associated with general linear differential operators and linear functionals were discussed. By constructing an appropriate reproducing kernel Hilbert space framework, the proposed splines were expressed as minimum norm problems. Thus the expression and interpolation error of the smoothing spline were obtained via reproducing kernel. Based on this, a new method for computing polynomial smoothing splines was presented.