In this paper, we propose a fully discrete soft thresholding trigonometric polynomial approximation on $[-\pi,\pi],$ named Lasso trigonometric interpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation under the same conditions of classical trigonometric interpolation on an equidistant grid. Lasso trigonometric interpolation is sparse and meanwhile it is an efficient tool to deal with noisy data. We theoretically analyze Lasso trigonometric interpolation for continuous periodic function. The principal results show that the $L_2$ error bound of Lasso trigonometric interpolation is less than that of classical trigonometric interpolation, which improved the robustness of trigonometric interpolation. This paper also presents numerical results on Lasso trigonometric interpolation on $[-\pi,\pi]$, with or without the presence of data errors.

翻译：本文提出一种在$[-\pi,\pi]$上的全离散软阈值三角多项式逼近方法，命名为Lasso三角插值。该逼近是在等距网格上，在经典三角插值的相同条件下进行$\ell_1$正则化离散最小二乘逼近。Lasso三角插值具有稀疏性，同时是处理含噪数据的有效工具。我们从理论上分析了针对连续周期函数的Lasso三角插值。主要结果表明，Lasso三角插值的$L_2$误差界小于经典三角插值，从而提高了三角插值的鲁棒性。本文还给出了$[-\pi,\pi]$上Lasso三角插值在存在或不存在数据误差情况下的数值结果。