Tikhonov regularization for polynomial approximation problems in Gauss quadrature points

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.

Automated summary

This paper introduces the application of Tikhonov regularization in least squares approximation using orthonormal polynomials to handle noisy data. By utilizing Gauss quadrature points as nodes, coefficients of the approximation polynomial are derived and error bounds are provided. Tikhonov regularization is shown to reduce the operator norm and error term related to noise level by introducing a correction factor.