2024/03/29 by An, Congpei, Cai, Mou
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2403.19927
This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific cases. With the aid of the de la Vallée-Poussin approximation, we derive a uniform error bound and a concrete L2 error bound. These error estimates demonstrate the effectiveness of Tikhonov regularization in the denoising process. A new regularity condition for the selection of regularization parameters is proposed. We investigate three strategies for choosing regularization parameters: Morozov's discrepancy principle, the L-curve, and generalized cross-validation, by explicitly combining these error bounds of the approximating trigonometric polynomial. We show that Morozov's discrepancy principle satisfies the proposed regularity condition, while the other two methods do not. Finally, numerical examples are provided to illustrate how the aforementioned methodologies, when applied with well-chosen parameters, can significantly improve the quality of approximation.