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 derive $L_2$ errors in theoretical analysis for hybrid hyperinterpolation to approximate continuous functions with noise data on sampling points. Numerical examples illustrate the theoretical results and show that well chosen regularization parameters can enhance the approximation quality over the unit-sphere and the union of disks.

翻译：本文在超插值假设下提出了一种基于$\ell^2_2+\ell_1$正则化离散最小二乘逼近方法，适用于一般区域，称为混合超插值。该方法结合了Lasso与滤波超插值技术，通过软阈值算子与滤波函数对基于高阶求积规则、针对给定连续函数在某一正交基下近似得到的傅里叶系数进行收缩处理。当正则化参数与滤波函数选取适当时，混合超插值能够继承二者的特性以处理含噪声数据。我们通过理论分析推导了混合超插值在采样点上逼近含噪声连续函数时的$L_2$误差。数值算例验证了理论结果，并表明在单位球面及圆盘并集区域上，恰当选取的正则化参数能够有效提升逼近质量。