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 operator norm (Lebesgue constants) 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.

中文翻译：

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高斯正交点多项式逼近问题的 Tikhonov 正则化

本文涉及通过正交多项式将 Tikhonov 正则化引入 $[-1,1]$ 上的最小二乘逼近方案，以处理噪声数据。该方案包括作为特殊情况的插值和超插值。使用高斯正交点作为节点，近似多项式相对于给定基的系数以入口闭合形式导出。在插值条件下，通过在两种经典重心公式中仅引入乘法校正因子，将正则化逼近问题的解改写为两种重心插值公式的形式。导出 $L_2$ 误差界限和统一误差界限，提供了类似的信息，即 Tikhonov 正则化能够通过乘以小于 1 的校正因子来减少算子范数（勒贝格常数）和与噪声水平相关的误差项。数值示例显示了当数据嘈杂或数据规模相对较小时 Tikhonov 正则化的好处。