论文标题
在使用冗余集和帧的功能的自适应光谱近似上
On the adaptive spectral approximation of functions using redundant sets and frames
论文作者
论文摘要
具有光谱基础的平滑函数的近似通常会导致迅速的衰减系数，其中衰减速率取决于功能的平滑度，反之亦然。一旦达到阈值，可以通过截断系数来确定近似值中的最佳自由度数量。基于冗余集和帧的最新近似方案将光谱近似值扩展到在不规则几何形状和某些非平滑函数上定义的功能的适用性。但是，由于其固有的冗余性，即使对于非常平滑的功能，帧近似中的膨胀系数也不一定会衰减。在本文中，我们强调了平滑度和系数衰减之间缺乏等效性，我们探索了确定此类冗余近似值的最佳自由度的方法。
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.
