在任意域上使用傅里叶扩展框架进行函数逼近
Function approximation on arbitrary domains using Fourier extension frames
AI总结：
本文提出一种在任意二维域上以O(N² log²N)复杂度计算傅里叶扩展框架逼近的算法，利用 plunge 区域现象提升计算效率，适用于具有分形边界的域。
摘要：
Fourier extension 是一种将任意有界域上的函数用经典傅里叶级数在包围盒上进行逼近的近似方法。在较小的域上，傅里叶级数表现出冗余性，其数学结构更接近于框架而非基。利用函数在域内点上的评估进行此类框架的逼近并不容易，但可通过离散最小二乘逼近实现。对应的系统极其病态，由于框架的冗余性，但通过正则化SVD求解可达到非常高的（几乎谱）精度。然而，该计算仍需O(N³)次操作。本文描述了一种在一般二维域上以仅O(N² log²N)次操作计算此类傅里叶扩展框架逼近的算法。对于更简单的张量积域，复杂度降低至O(N log²N)。该算法利用了时间-频率局部化算子分析中的现象称为 plunge 区域，其在本文中表现为最小二乘矩阵奇异值的突然下降。已知在一维问题中 plunge 区域的大小大约为O(log N)。本文证明，在完全离散情况下，对于大多数二维域，plunge 区域的大小大约为O(N log N)，证明了一个与连续问题相关结果的离散等价性。复杂度估计依赖于域边界的Minkowski或盒维数，因此对于具有分形形状的域，其复杂度将大于O(N log N)。
英文摘要：
Fourier extension is an approximation scheme in which a function on an arbitary bounded domain is approximated using a classical Fourier series on a bounding box. On the smaller domain the Fourier series exhibits redundancy, and it has the mathematical structure of a frame rather than a basis. It is not trivial to construct approximations in this frame using function evaluations in points that belong to the domain only, but one way to do so is through a discrete least squares approximation. The corresponding system is extremely ill-conditioned, due to the redundancy in the frame, yet its solution via a regularized SVD is known to be accurate to very high (and nearly spectral) precision. Still, this computation requires ${\mathcal O}(N^3)$ operations. In this paper we describe an algorithm to compute such Fourier extension frame approximations in only ${\mathcal O}(N^2 \log^2 N)$ operations for general 2D domains. The cost improves to ${\mathcal O}(N \log^2N)$ operations for simpler tensor-product domains. The algorithm exploits a phenomenon called the plunge region in the analysis of time-frequency localization operators, which manifests itself here as a sudden drop in the singular values of the least squares matrix. It is known that the size of the plunge region scales like ${\mathcal O}(\log N)$ in one dimensional problems. In this paper we show that for most 2D domains in the fully discrete case the plunge region scales like ${\mathcal O}(N \log N)$, proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. The complexity estimate depends on the Minkowski or box-counting dimension of the domain boundary, and as such it is larger than ${\mathcal O}(N \log N)$ for domains with fractal shape.
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