We present an $\ell^2_2+\ell_1$-regularized discrete least squares approximation over general regions under assumptions of hyperinterpolation, named hybrid hyperinterpolation. Hybrid hyperinterpolation, using a soft thresholding operator as well a filter function to shrink the Fourier coefficients approximated by a high-order quadrature rule of a given continuous function with respect to some orthonormal basis, is a combination of lasso and filtered hyperinterpolations. Hybrid hyperinterpolation inherits features of them to deal with noisy data once the regularization parameters and filter function are chosen well. We not only provide $L_2$ errors in theoretical analysis for hybrid hyperinterpolation to approximate continuous functions with noise and noise-free, but also decompose $L_2$ errors into three exact computed terms with the aid of a prior regularization parameter choices rule. This rule, making fully use of coefficients of hyperinterpolation to choose a regularization parameter, reveals that $L_2$ errors for hybrid hyperinterpolation sharply decline and then slowly increase when the sparsity of coefficients ranges from one to large values. Numerical examples show the enhanced performance of hybrid hyperinterpolation when regularization parameters and noise vary. Theoretical $L_2$ errors bounds are verified in numerical examples on the interval, the unit-disk, the unit-sphere and the unit-cube, the union of disks.

翻译：我们提出了一种在一般区域上基于超插值假设的ℓ2²+ℓ1正则化离散最小二乘逼近方法，称为混合超插值。混合超插值采用软阈值算子与滤波器函数，对由给定连续函数在正交基下的高阶求积规则逼近的傅里叶系数进行收缩，是LASSO方法与滤波超插值的结合。当正则化参数和滤波器函数选择恰当时，混合超插值继承了这两种方法处理含噪数据的特性。我们不仅在理论分析中提供了混合超插值逼近连续函数（含噪声与无噪声情况）的L2误差，还借助先验正则化参数选择准则将L2误差分解为三个精确计算项。该准则充分利用超插值系数来选择正则化参数，揭示了当系数稀疏度从1增至较大值时，混合超插值的L2误差先急剧下降后缓慢上升的规律。数值算例表明，在不同正则化参数和噪声条件下，混合超插值均展现出增强性能。在区间、单位圆盘、单位球面、单位立方体及圆盘并集上的数值算例验证了理论L2误差界的正确性。