This paper develops a fully discrete soft thresholding polynomial approximation over a general region, named Lasso hyperinterpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation under the same conditions of hyperinterpolation. Lasso hyperinterpolation also uses a high-order quadrature rule to approximate the Fourier coefficients of a given continuous function with respect to some orthonormal basis, and then it obtains its coefficients by acting a soft threshold operator on all approximated Fourier coefficients. Lasso hyperinterpolation is not a discrete orthogonal projection, but it is an efficient tool to deal with noisy data. We theoretically analyze Lasso hyperinterpolation for continuous and smooth functions. The principal results are twofold: the norm of the Lasso hyperinterpolation operator is bounded independently of the polynomial degree, which is inherited from hyperinterpolation; and the $L_2$ error bound of Lasso hyperinterpolation is less than that of hyperinterpolation when the level of noise becomes large, which improves the robustness of hyperinterpolation. Explicit constructions and corresponding numerical examples of Lasso hyperinterpolation over intervals, discs, spheres, and cubes are given.
翻译:本文开发了一个全区域完全离散的软阈值多球近似值, 名为 Lasso 超内插。 这个近似值是 $\ ell_ 1 $ 1 常规化的离散最小正方形近似值, 与超内插条件相同 。 超极内插还使用高阶二次值规则, 以近似于某些正异性基点的四倍连续函数的四倍系数, 然后它通过对所有近似 Fleier 系数的软阈值操作员操作获得其系数。 激光超内插不是离散的或异的预测, 而是处理噪音数据的高效工具 。 我们从理论上分析激光超高极间插值, 用于连续和顺畅的函数 。 主要结果具有双重性: 激光超超内插操作器操作员的规范, 与超超超超多数值级的多数值; 激光超导的超值误差值比超超值的超值内插值差值差值, 且超度是超度的中间层。