针对使用冗余数据集和框架的函数采取自适应光谱近似(cs.NA)---用户7199428

基于光谱的平滑函数近似普遍会导致快速衰减的系数。该种情况下，衰变率依赖于函数的平滑度，反过来亦然。近似过程中最优化自由度的获取相对容易，一旦达到阈值，截断系数即可。近来的近似模型是基于冗余的集合和框架，使光谱近似进一步适用于基于不规则几何学的函数和特定的非平滑函数。但是，由于他们继承而来的冗余性，框架近似下的膨胀系数即使在非常平滑的函数上也不一定会衰减。在这篇论文中，我们强调了平滑度与系数衰减之间并不存在等值关系，并且我们研究出了一种方法用于这种冗余近似下的最优自由度获取。

原文题目：On the adaptive spectral approximation of functions using redundant sets and frames

原文：The approximation of smooth functions with a spectral basis typically leads to rapidly decaying coefficients where the rate of decay depends on the smoothness of the function and vice-versa. The optimal number of degrees of freedom in the approximation can be determined with relative ease by truncating the coefficients once a threshold is reached. Recent approximation schemes based on redundant sets and frames extend the applicability of spectral approximations to functions defined on irregular geometries and to certain non-smooth functions. However, due to their inherent redundancy, the expansion coefficients in frame approximations do not necessarily decay even for very smooth functions. In this paper, we highlight this lack of equivalence between smoothness and coefficient decay and we explore approaches to determine an optimal number of degrees of freedom for such redundant approximations.

原文作者：Vincent Coppé, Daan Huybrechs

原文地址：https://arxiv.org/abs/2004.11317