$d$维流形的双重傅里叶球面方法
A double Fourier sphere method for $d$-dimensional manifolds
AI总结：
本文提出一种广义双重傅里叶球面（DFS）方法，将定义在$d$维流形上的函数变换到同维环面进行傅里叶逼近，并证明其一致收敛性及Hölder连续函数的解析收敛速率。
摘要：
双重傅里叶球面（DFS）方法利用巧妙的技巧，将定义在单位球面上的函数变换到环面，随后用傅里叶级数逼近它，该级数可通过快速傅里叶变换高效计算。针对圆盘、球体和圆柱体上的逼近问题，也出现了类似的方法。在本文中，我们引入了一种适用于各种流形的广义DFS方法，包括上述所有情况以及更多情况，如旋转群。该方法将定义在流形上的函数变换到同维度的环面上。我们证明，变换后函数的傅里叶级数可以转换回流形上，并在流形上一致收敛于原函数。特别地，对于流形上的Hölder连续函数，我们获得了解析收敛速率。
英文摘要：
The double Fourier sphere (DFS) method uses a clever trick to transform a function defined on the unit sphere to the torus and subsequently approximate it by a Fourier series, which can be evaluated efficiently via fast Fourier transforms. Similar approaches have emerged for approximation problems on the disk, the ball, and the cylinder. In this paper, we introduce a generalized DFS method applicable to various manifolds, including all the above-mentioned cases and many more, such as the rotation group. This approach consists in transforming a function defined on a manifold to the torus of the same dimension. We show that the Fourier series of the transformed function can be transferred back to the manifold, where it converges uniformly to the original function. In particular, we obtain analytic convergence rates in case of Hölder-continuous functions on the manifold.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
