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.
翻译:本文涉及将Tikhonov 正规化纳入关于 $[1,1,1美元] 的最小正方位近似方案,以便通过正统多边多边货币处理吵闹的数据。这个方案包括作为特例的内插和超超内插。用高斯二次曲线点作为节点,近似多元货币相对于给定基点的系数以自入式封闭形式产生。在内部条件下,对正统近似问题的解决方案以两种巴里中心内插公式的形式重新写成,即两种巴里中心内插公式,即只对古典的巴里中心公式采用多倍校正系数。一个受困的L_2美元误差和一个统一的误差,提供类似信息,说明Tikhonov正规化能够降低操作员规范(Lebesgue stand)和与噪音水平有关的错误术语,两者均乘以一个小于一个的校正系数。数字实例显示,当数据变暖化或数据大小较小时,Tikhoonov会有好处。