The recent results presented in arXiv:2202.05608 have led to significant developments in achieving stable approximations of Helmholtz solutions by plane wave superposition. The study shows that the numerical instability and ill-conditioning inherent in plane wave-based Trefftz methods can be effectively overcome with regularization techniques, provided there exist accurate approximations in the form of expansions with bounded coefficients. Whenever the target solution contains high Fourier modes, propagative plane waves fail to yield stable approximations due to the exponential growth of the expansion coefficients. Conversely, evanescent plane waves, whose modal content covers high Fourier regimes, are able to provide both accurate and stable results. The developed numerical approach, which involves constructing evanescent plane wave approximation sets by sampling the parametric domain according to a probability density function, results in substantial improvements when compared to conventional propagative plane wave schemes. The following work extends this research to the three-dimensional setting, confirming the achieved results and introducing new ones. By generalizing the 3D Jacobi$-$Anger identity to complex-valued directions, we show that any Helmholtz solution in a ball can be represented as a continuous superposition of evanescent plane waves. This representation extends the classical Herglotz one and provides a relevant stability result that cannot be achieved with the use of propagative waves alone. The proposed numerical recipes have been tailored for the 3D setting and extended with new sampling strategies involving extremal systems of points. These methods are tested by numerical experiments, showing the desired accuracy and bounded-coefficient stability, in line with the two-dimensional case.

翻译：arXiv:2202.05608中提出的最新结果在通过平面波叠加实现亥姆霍兹解的稳定近似方面取得了重大进展。研究表明，只要存在以有界系数展开形式的精确近似，基于平面波的Trefftz方法中固有的数值不稳定性和病态性便可通过正则化技术有效克服。当目标解包含高频傅里叶模态时，传播型平面波会因展开系数的指数增长而无法提供稳定近似。相反，其模态内容覆盖高频区域的倏逝平面波能够同时提供精确且稳定的结果。所提出的数值方法通过根据概率密度函数对参数域进行采样来构造倏逝平面波近似集，与传统的传播型平面波方案相比实现了显著改进。本研究将该工作拓展至三维场景，既验证了已有结论又引入新成果。通过将三维Jacobi$-$Anger恒等式推广至复值方向，我们证明球内任意亥姆霍兹解均可表示为倏逝平面波的连续叠加。该表示形式扩展了经典Herglotz表示，并提供了纯传播型波无法实现的关键稳定性结果。所提出的数值方案针对三维场景进行了定制，并引入了涉及极值点系统的新采样策略。数值实验表明，这些方法与二维情况一致，达到了所需的精度和有界系数稳定性。