AI 中文总结
该研究针对具有可求长Jordan边界的平面区域上解析函数的面积积分，基于柯西 - 格林恒等式，利用AAA算法计算有理函数逼近边界值，构造求积规则，有对偶解读，数值例子表现良好，还追踪了相关解析骨架。
AI 中文摘要
我们给出了一种用于在具有可求长Jordan边界的平面区域上对解析函数的面积积分进行求积规则的数值构造。出发点是柯西 - 格林恒等式。给定一个权重w，我们选择一个\(\bar\partial\) - 反导数W，并将面积积分简化为一个涉及W边界值的围道积分。然后通过AAA算法计算的具有自由极点的有理函数来逼近这些值。区域内的极点成为求积节点，相应的留数成为权重，边界残差通过后验估计控制误差，一旦连续边界残差有界则该估计严格。相同规则有对偶解读，即作为被积函数的有理插值的精确积分，是一维插值观点的面积类似物。数值例子以机器精度恢复了圆盘均值规则和椭圆的焦点段规则，再现了具有分离和汇合节点的求积域的精确有限求积恒等式，并仅从边界数据评估对数和柯西体积势。内部极点追踪我们初步认定为势理论母体的解析骨架以及未被施加而出现的像点；对于正方形，观察到的收敛是根指数型的。
英文摘要
We present a numerical construction of cubature rules for area integrals of analytic functions over planar domains with rectifiable Jordan boundary. The starting point is the Cauchy--Green identity. Given a weight $w$, we choose a $\bar\partial$-antiderivative $W$ and reduce the area integral to a contour integral involving the boundary values of $W$. These values are then approximated by a rational function with free poles, computed by the AAA algorithm. The poles inside the domain become cubature nodes, the corresponding residues become weights, and the boundary residual controls the error through an a posteriori estimate, rigorous once the continuous boundary residual is bounded. The same rule admits a dual reading, as the exact integral of a rational interpolant to the integrand, the area analogue of the one-dimensional interpolatory viewpoint. The numerical examples recover the disk mean-value rule and the focal-segment rule of the ellipse to machine precision, reproduce the exact finite quadrature identities of quadrature domains with both separated and confluent nodes, and evaluate logarithmic and Cauchy volume potentials from boundary data alone. The interior poles trace analytic skeletons that we identify tentatively with the mother bodies of potential theory, along with image points that appear without being imposed; for the square the observed convergence is root-exponential.
Comments14 pages, 7 figures, 3 tables
