We propose a fully discrete hard thresholding polynomial approximation over a general region, named hard thresholding hyperinterpolation (HTH). This approximation is a weighted $\ell_0$-regularized discrete least squares approximation under the same conditions of hyperinterpolation. Given an orthonormal basis of a polynomial space of total-degree not exceeding $L$ and in view of exactness of a quadrature formula at degree $2L$, HTH approximates the Fourier coefficients of a continuous function and obtains its coefficients by acting a hard thresholding operator on all approximated Fourier coefficients. HTH is an efficient tool to deal with noisy data because of the basis element selection ability. The main results of HTH for continuous and smooth functions are twofold: the $L_2$ norm of HTH operator is bounded independently of the polynomial degree; and the $L_2$ error bound of HTH is greater than that of hyperinterpolation but HTH performs well in denoising. We conclude with some numerical experiments to demonstrate the denoising ability of HTH over intervals, discs, spheres, spherical triangles and cubes.
翻译:我们建议对一般区域采用完全离散的硬阈值多球形近似值,称为硬阈值超内插(HTH)。这一近似值是一种在超内插相同条件下的加权 $ ell_0$0$ 常规离散最小正方形近似值。鉴于总度不超过$L$的多元空间的正统性基础,并鉴于2L$的四方公式的精确度,HTH接近连续函数的四倍系数,并通过对所有四倍系数的近似临界值采取硬阈值操作器来获取系数。HTH是一个处理噪音数据的有效工具,因为基本元素选择能力。 HTH是连续和顺畅功能的主要结果具有双重性:HTH操作员的值为$L_2$标准与多元度无关;HTH约束的$L_2美元差值大于超度,但HTH在解析中表现良好。我们最后进行了一些数字实验,以显示HTH在隔间、盘、立方体和界域上的解析能力。