In a previous paper (Adcock and Huybrechs in SIAM Rev 61(3):443–473, 2019) we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but ill-conditioning often prevents the numerical computation of best approximations. We showed that, in spite of said ill-conditioning, approximations with regularization may still provide accuracy up to order \(\sqrt{\epsilon }\), where \(\epsilon \) is a small truncation threshold. When using frames, i.e. complete systems that are generally redundant but which provide infinite representations with coefficients of bounded norm, this accuracy can actually be achieved for all functions in a space. Here, we generalize that setting in two ways. We assume information or samples from f from a wide class of linear operators acting on f, rather than inner products associated with the best approximation projection. This enables the analysis of fully discrete approximations based, for instance, on function values only. Next, we allow oversampling, leading to least-squares approximations. We show that this leads to much improved accuracy on the order of \(\epsilon \) rather than \(\sqrt{\epsilon }\). Overall, we demonstrate that numerical function approximation using redundant representations may lead to highly accurate approximations in spite of having to solve ill-conditioned systems of equations.

在先前的论文中（Adcock和Huybrechs在SIAM Rev 61（3）：443–473，2019中），我们描述了使用冗余集和框架的函数的数值逼近。与使用基础相比，函数表示中的冗余提供了极大的灵活性，但是条件不佳通常会阻止最佳近似值的数值计算。我们表明，尽管存在不适，但使用正则化的近似值仍可以提供高达\（\ sqrt {\ epsilon} \）的精度，其中\（\ epsilon \）是一个小的截断阈值。当使用框架时，即通常是冗余的完整系统，但其提供具有有界范数系数的无限表示时，实际上可以为空间中的所有功能实现此精度。在这里，我们以两种方式概括该设置。我们假设作用于f的各种线性算子来自f的信息或样本，而不是与最佳近似投影相关的内积。这样就可以仅基于函数值来分析完全离散的近似值。接下来，我们允许过采样，从而得出最小二乘近似。我们证明，这导致\（\ epsilon \）而不是\（\ sqrt {\ epsilon} \）的精度大大提高。。总的来说，我们证明了使用冗余表示的数值函数逼近可能会导致高度精确的逼近，尽管必须解决病态方程组。