论文标题
高斯正交点多项式近似问题的Tikhonov正则化
Tikhonov regularization for polynomial approximation problems in Gauss quadrature points
论文作者
论文摘要
本文涉及将Tikhonov正则化在正式多项式上以$ [-1,1] $上的最小二乘近似方案的引入，以处理嘈杂的数据。该方案包括插值和过度中断为特殊情况。用作为节点使用的高斯正交点，相对于给定的基础，近似多项式的系数以入口封闭形式得出。在插值条件下，通过仅将乘法校正因子引入两个经典的barycentric公式，以两种Barycentric插值公式的形式重写。 $ L_2 $错误绑定和统一误差绑定得出，提供了类似的信息，即Tikhonov正则化能够减少运算符规范（Lebesgue常数）和与噪声水平相关的误差项，均通过将校正因子乘以小于一个的校正因子。数值示例表明，当数据嘈杂或数据大小相对较小时，Tikhonov正则化的好处。
This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on $[-1,1]$ by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and hyperinterpolation as special cases. With Gauss quadrature points employed as nodes, coefficients of the approximation polynomial with respect to given basis are derived in an entry-wise closed form. Under interpolatory conditions, the solution to the regularized approximation problem is rewritten in forms of two kinds of barycentric interpolation formulae, by introducing only a multiplicative correction factor into both classical barycentric formulae. An $L_2$ error bound and a uniform error bound are derived, providing similar information that Tikhonov regularization is able to reduce the operator norm (Lebesgue constant) and the error term related to the level of noise, both by multiplying a correction factor which is less than one. Numerical examples show the benefits of Tikhonov regularization when data is noisy or data size is relatively small.
