AI 中文总结
本文提出算子框架构造无散无旋矩阵值核，降低势方法正则性要求，完成核插值问题的精确误差分析与稳定性分析，数值实验验证了理论结果。
AI 中文摘要
矩阵值核为从散乱数据近似向量场提供了灵活框架，尤其适用于必须保持无散或无旋等结构约束的场景。经典的基于势的构造可自然满足这些约束，但通常要求生成标量函数具有较高光滑性。本文提出一种基于积分与微分算子的算子框架，用于构造无散和无旋矩阵值核，大幅降低了势方法的正则性要求。利用维数行走技术，证明所得原生空间与合适的向量值 Sobolev 空间范数等价。本文另一主要贡献是对相应的矩阵值核插值问题进行了精确误差分析：推导了允许目标场具有分数阶正则性的直接 Sobolev 误差估计，建立了关联核试验空间的 Bernstein 型不等式，这些结果构成了完整的逆定理；还通过证明插值矩阵最小特征值的下界研究了稳定性，最后通过数值实验验证了理论结果。
英文摘要
Matrix-valued kernels provide a flexible framework for approximating vector fields from scattered data, especially when structural constraints such as divergence-free or curl-free conditions must be preserved. Classical potential-based constructions enforce these constraints naturally, but they typically require the generating scalar function to possess relatively high smoothness. We develop an operator-based framework for constructing div-free and curl-free matrix-valued kernels using integral and differential operators, which substantially relaxes the regularity requirements of the potential approach. Using dimension-walking techniques, we show that the resulting native spaces are norm-equivalent to appropriate vector-valued Sobolev spaces. Another main contribution of the paper is a sharp error analysis for the corresponding kernel matrix-valued interpolation problem. We derive direct Sobolev error estimates that allow fractional regularity of the target field, and we establish Bernstein-type inequalities for the associated kernel trial spaces. These results lead to a complete inverse theorem. We also investigate stability by proving lower bounds for the smallest eigenvalues of the interpolation matrices. Numerical experiments are included to verify the theoretical results.
