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 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：本文提出了一种在一般区域上基于超插值假设的$\ell^2_2+\ell_1$正则化离散最小二乘逼近方法，称为混合超插值。混合超插值利用软阈值算子和滤波函数，通过高阶求积规则对给定连续函数的傅里叶系数（相对于某个标准正交基）进行逼近并收缩，是Lasso与滤波超插值的结合。一旦正则化参数和滤波函数选择得当，混合超插值能继承这两种方法的特性以处理含噪数据。我们不仅在理论上分析了混合超插值逼近含噪和无噪连续函数的$L_2$误差，还借助先验正则化参数选择规则将$L_2$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来选择正则化参数，揭示了当系数稀疏度从1变化到较大值时，混合超插值的$L_2$误差先急剧下降后缓慢增加。数值算例展示了混合超插值在正则化参数和噪声变化时的增强性能。在区间、单位圆盘、单位球面、单位立方体以及圆盘并集上的数值算例中验证了理论$L_2$误差界。