莫比乌斯变换梯形法则
Möbius-Transformed Trapezoidal Rule
AI总结：
本文提出一种基于莫比乌斯变换的梯形法则，用于在加权Sobolev空间中实现最优收敛速度的数值积分方法，并扩展了其在周期函数积分和多变量积分中的应用。
摘要：
我们通过将梯形法则与一种将单位圆映射到实线的莫比乌斯变换相结合，研究数值积分。我们证明，如果被积函数生活在具有权重的加权Sobolev空间中，该权重仅被假设为一个正的Schwartz函数，且在无穷远处单调递减到零，则所得到的变换梯形法则能获得最优的收敛速度。我们的算法仅需能够评估所选节点处的权重，而不需要从由权重定义的概率测度中采样，也不需要了解其导数信息。特别是，我们证明莫比乌斯变换作为实线与单位圆之间的变量变换，将加权Sobolev空间中的函数映射到具有相同光滑度的周期Sobolev空间。由于已有许多关于积分和逼近周期函数的结果，我们还描述了莫比乌斯变换梯形法则的几种扩展，包括通过三角插值进行函数逼近、使用随机算法进行积分以及多变量积分。
英文摘要：
We study numerical integration by combining the trapezoidal rule with a Möbius transformation that maps the unit circle onto the real line. We prove that the resulting transformed trapezoidal rule attains the optimal rate of convergence if the integrand function lives in a weighted Sobolev space with a weight that is only assumed to be a positive Schwartz function decaying monotonically to zero close to infinity. Our algorithm only requires the ability to evaluate the weight at the selected nodes, and it does not require sampling from a probability measure defined by the weight nor information on its derivatives. In particular, we show that the Möbius transformation, as a change of variables between the real line and the unit circle, sends a function in the weighted Sobolev space to a periodic Sobolev space with the same smoothness. Since there are various results available for integrating and approximating periodic functions, we also describe several extensions of the Möbius-transformed trapezoidal rule, including function approximation via trigonometric interpolation, integration with randomized algorithms, and multivariate integration.
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