We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show a general result relating boundedness, solvability, and convergence that can be used to find eigenvalues. The method works by extending a solution to a surface PDE into a box-shaped domain so that the differential operators of the extended function agree with the surface differential operators, as in the Closest Point Method. This differs from approaches that require a basis for the surface of interest, which may not be available. Numerical experiments are also provided, demonstrating super-algebraic convergence. Current high-order methods for surface PDEs are often limited to a small class of surfaces or use radial basis functions (RBFs). Our approach offers certain advantages related to conditioning, generality, and ease of implementation. The method is meshfree and works on arbitrary surfaces (closed or non-closed) defined by point clouds with minimal conditions.

翻译：本文分析并测试了通过最小化希尔伯特空间范数的傅里叶延拓方法，用于求解曲面上的偏微分方程。我们证明了该方法具有任意高阶精度，并给出了一个有界性、可解性与收敛性之间的一般性关系定理，该定理可用于特征值求解。该方法的核心思想是将曲面偏微分方程的解延拓至一个盒状区域，使得延拓函数的微分算子与曲面微分算子保持一致，这与最近点方法的思想类似。该方法不同于那些需要已知曲面基函数的传统方法（此类基函数往往难以获得）。数值实验结果表明该方法具有超代数收敛性。当前求解曲面偏微分方程的高阶方法通常局限于特定类型的曲面，或需借助径向基函数。本文方法在条件数、普适性及实现简便性方面具有优势。该方法为无网格方法，适用于由点云定义的任意曲面（封闭或非封闭），且对点云分布要求极低。