标题： 有限三维样品的磁矩：从大圆盘上的平面测量中渐近恢复 显示英文标题

标题： Magnetisation moment of a bounded 3D sample: asymptotic recovery from planar measurements on a large disk

摘要： 我们考虑从磁场所的部分数据中重建样品的整体磁化矢量（净磁矩）的问题。 具体来说，受一个具体的实验装置的启发，我们处理一种情况，即磁场在靠近样品的平面的一部分上进行测量，并且仅可获得场的一个（垂直于平面）分量。 在假设测量区域是一个足够大的圆盘（位于样品上方的水平平面上）的前提下，我们得到了净磁矩矢量各分量的一组估计，其精度随着测量圆盘半径的增加而逐步提高。 与我们之前的初步结果相比，现在的渐近公式得到了严格的证明，并推导出了更高阶的估计。 此外，所提出的方法基于傅里叶域中的适当分解以及振荡积分的估计（涉及小参数和大参数），阐明了任意阶渐近估计的推导，这一可能性之前是不清楚的。 所得结果通过数值模拟进行了说明，并讨论了其对噪声的鲁棒性。 所提出的方法应适用于其他具有平面测量的磁性和重力问题。 显示英文摘要

摘要： We consider the problem of reconstruction of the overall magnetisation vector (net moment) of a sample from partial data of the magnetic field. Namely, motivated by a concrete experimental set-up, we deal with a situation when the magnetic field is measured on a portion of the plane in vicinity of the sample and only one (normal to the plane) component of the field is available. Under assumption that the measurement area is a sufficiently large disk (lying in a horizontal plane above the sample), we obtain a set of estimates for the components of the net moment vector with the accuracy which improves asymptotically with the increase of the measurement disk radius. Compared to our previous preliminary results, the asymptotic formulas are now rigorously justified and higher-order estimates are derived. Moreover, the presented approach, based on an appropriate splitting in the Fourier domain and estimates of oscillatory integrals (involving both small and large parameters), elucidates the derivation of asymptotic estimates of an arbitrary order, a possibility that was previously unclear. The obtained results are illustrated numerically and their robustness with respect to noise is discussed. The proposed methodology should be applicable to other magnetic and gravimetric problems with planar measurements.