Bessel变换的一个下界

A Lower Bound of Bessel Inversion

Bessel逆问题在物理、化学和工程学等诸多领域有重要应用.解决线性逆问题的传统方法不适合处理具有奇异性曲线边缘的二元函数.鉴于切波对这一类函数的最优表示能力,相关文献采用切波方法研究Bessel逆问题,构造了目标函数的切波域值估计器,得到了它在函数空间V中积分均方差收敛阶的上界.在此基础上利用统计理论给出其最小最大风险的一个下界,证明了在估计Bessel逆问题时此估计器是最优的.

The inverse problem of Bessel transform with additive noise plays important roles in scientific settings ranging from physics,chemistry to engineering.Traditional linear methods for solving such inverse problem behave poorly in the presence of edges.In view that shearlets provides essentially optimal sparse approximation for 2D functions containing curve,Hu and Liu apply shearlets to the the inverse problem of Bessel transform.They construct the shearlet thresholding estimator,and obtain an upper bound of the the mean square error convergence rate over the space V.This paper is devoted to show a lower bound to the minimax mean square error for that class of functions.It turns out that the shearlet thresholding estimator attains the optimal mean square error convergence rate,ignoring log factor.