论文标题
从傅立叶数据中重建分段平滑的多元函数
Reconstruction of piecewise-smooth multivariate functions from Fourier data
论文作者
论文摘要
在某些应用中，一个人有兴趣从其傅立叶系列系数中重建功能$ f $。问题在于，如果该函数是非周期性的，则傅立叶级数会缓慢收敛，或者是不平滑的。在本文中，我们建议使用类似帕德的方法将高阶近似值推导为$ f $的方法。也就是说，通过将近似值的一些傅立叶系数安装在给定的傅立叶系数$ f $的情况下。鉴于$ \ mathbb {r}^d $在矩形域上函数的傅立叶级数系数，假设函数是分段平滑的，我们通过分段高阶的高阶样条函数近似该函数。首先，确定了函数的奇异性结构。例如，在2-D情况下，我们发现与$ f $平滑段之间的曲线相对于曲线的近似值很高。其次，同时我们发现了$ f $的所有不同部分的近似值。我们首先为1-D情况开发和演示高精度算法，然后使用此算法来升级到多维情况。
In some applications, one is interested in reconstructing a function $f$ from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving high order approximation to $f$ using a Padé-like method. Namely, by fitting some Fourier coefficients of the approximant to the given Fourier coefficients of $f$. Given the Fourier series coefficients of a function on a rectangular domain in $\mathbb{R}^d$, assuming the function is piecewise smooth, we approximate the function by piecewise high order spline functions. First, the singularity structure of the function is identified. For example in the 2-D case, we find high accuracy approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously we find the approximations of all the different segments of $f$. We start by developing and demonstrating a high accuracy algorithm for the 1-D case, and we use this algorithm to step up to the multidimensional case.
