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 and 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 parameter and the 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 decrease 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ℓ²_2+ℓ₁正则化离散最小二乘逼近方法，命名为混合超插值。混合超插值通过软阈值算子和滤波函数，对由给定连续函数关于某正交基的高阶求积规则逼近得到的傅里叶系数进行收缩，是Lasso与滤波超插值的结合。当正则化参数和滤波函数选取恰当时，混合超插值继承了二者处理含噪数据的特性。我们不仅从理论上分析了混合超插值逼近含噪与无噪连续函数的L₂误差，还借助一个先验正则化参数选择规则，将L₂误差分解为三个精确可计算的项。该规则充分利用超插值系数选择正则化参数，揭示了当系数的稀疏性从小值增加到大值时，混合超插值的L₂误差先急剧下降后缓慢上升的规律。数值算例展示了正则化参数与噪声变化时混合超插值的增强性能。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。