AI 中文总结
本文针对球面积分方程，通过球谐函数分析Sloan迭代的超收敛与混叠饱和，推导了分离谱尾与混叠贡献的严格界，明确了过求积对恢复收敛率的阈值，并用数值实验验证了结果。
AI 中文摘要
Sloan迭代可提升Galerkin法与退化核近似对第二类积分方程的收敛阶，经求积离散后，二者分别变为离散Galerkin法与乘积积分Nyström法。本文针对球面上的带状积分方程，通过将Sloan误差恒等式用球谐函数表示，明确回答了求积离散能保留多少这种收敛阶提升的问题。无求积时，Sloan迭代可充分利用积分算子的平滑性；求积会将未解析信息混叠到低频模式中，破坏该增益，使进一步迭代无法提升渐近收敛率，形成四种方法统一的尾-混叠描述。对于正权多项式精确求积及精确代数阶乘子，本文推导了分离谱尾与混叠贡献的严格双侧最坏情况界，二者的平衡产生了依赖深度的恢复-饱和阈值：足够的过求积可恢复无求积时的收敛率，低于阈值时混叠决定严格收敛阶。本文还借助Marcinkiewicz-Zygmund稳定性与Gram校正最小二乘法，将分析扩展至多项式精确性之外的情况，数值实验验证了两种机制。
英文摘要
Sloan iteration raises the convergence order of Galerkin and degenerate-kernel approximations to second-kind integral equations. After quadrature discretization, these become a discrete Galerkin method and a product-integration Nyström method, respectively. How much of this improvement survives quadrature discretization? For zonal integral equations on the sphere, we give a sharp answer by expressing the Sloan error identity in terms of spherical harmonics. In the absence of quadrature, Sloan iteration fully exploits the smoothing of the integral operator. Quadrature can destroy this gain by aliasing unresolved information into low-frequency modes, where further iteration no longer improves the asymptotic rate. This yields a unified tail--aliasing description of these four methods. For positive-weight polynomially exact quadrature and multipliers of exact algebraic order, we derive sharp two-sided worst-case bounds that separate the spectral-tail and aliasing contributions. Their balance yields a depth-dependent recovery--saturation threshold: sufficient overintegration recovers the quadrature-free rate, while below the threshold aliasing determines the sharp order. We also extend the analysis beyond polynomial exactness using Marcinkiewicz--Zygmund stability and Gram-corrected least squares. Numerical experiments illustrate both regimes.
Comments18 pages, 2 figures
