AI 中文总结
研究利用柱面调和函数对亥姆霍兹方程进行空间相关定量龙格逼近，推导了三圆配置及一般几何设置下的相关指标，为展开系数提供范数估计，有助于构造谱精确数值逼近。
AI 中文摘要
源于全纯函数的复逼近，椭圆方程的龙格逼近自被拉克斯、马尔格兰奇提出后，已发展成为反问题乃至基于学习的数值方法的基本工具，鲁兰德和萨洛进一步建立了定量表征。需注意龙格逼近是不适定的。在数值分析中，需要显式定量估计来表征近似值增长对原解向外延拓距离的依赖性。本文利用柱面调和函数研究亥姆霍兹方程的空间相关定量龙格逼近，考虑了内、外边界值问题。明确推导了三圆配置的相关指标，并得到了一般几何设置下的渐近指标。所得结果为展开系数提供了范数估计，这对正则化方法的实施至关重要。此外，所建立的界使得能够构造亥姆霍兹方程解的谱精确数值逼近。
英文摘要
Originating from the complex approximation of holomorphic functions, the Runge approximation for elliptic equations has evolved into a fundamental tool for inverse problems and even learning-based numerical methods since its proposition by Lax, Malgrange, with quantitative characterizations further established by Rüland and Salo. It should be remarked here that Runge approximation is ill-posed. In numerical analysis, explicit quantitative estimates are required to characterize the dependence of the approximant's growth on the outward continuation distance of the original solution. This paper investigates the spatial dependent quantitative Runge approximation for the Helmholtz equation using cylindrical harmonics, considering both interior and exterior boundary value problems. We explicitly derive the relevant indices for the three-circle configuration and obtain asymptotic indices for general geometric settings. The derived results provide norm estimates for the expansion coefficients, which are crucial for the implementation of regularization methods. Furthermore, the established bounds enable the construction of spectrally accurate numerical approximations for solutions to the Helmholtz equation.
