标题： 通过径向平移依赖和经典FFT进行$\mathbb{Z}^2\backslash SE(2)$上的快速卷积 显示英文标题

标题： Fast Convolutions on $\mathbb{Z}^2\backslash SE(2)$ via Radial Translational Dependence and Classical FFT

摘要： 令 $\mathbb{Z}^2\backslash SE(2)$ 表示由非阿贝尔平面运动群 $SE(2)$ 中正交格子 $\mathbb{Z}^2$ 的平移等距构形构成的子群的右陪集空间。 本文提出了一种快速准确的数值方法，用于逼近 $\mathbb{Z}^2\backslash SE(2)$ 上的函数。 我们利用有限傅里叶系数，讨论右陪集空间 $\mathbb{Z}^2\backslash SE(2)$ 上的函数有限傅里叶级数。 研究了有限傅里叶系数的收敛性/误差分析。 建立了有限傅里叶系数收敛于傅里叶系数的条件。 讨论了有限变换的矩阵形式。 本文研究了离散方法的实现，该方法用于计算具有平移径向函数的$SE(2)$-卷积的数值近似值。 本文最后讨论了该数值方案在开发用于近似具有平移径向函数的多重卷积的快速算法方面的能力。 显示英文摘要

摘要： Let $\mathbb{Z}^2\backslash SE(2)$ denote the right coset space of the subgroup consisting of translational isometries of the orthogonal lattice $\mathbb{Z}^2$ in the non-Abelian group of planar motions $SE(2)$. This paper develops a fast and accurate numerical scheme for approximation of functions on $\mathbb{Z}^2\backslash SE(2)$. We address finite Fourier series of functions on the right coset space $\mathbb{Z}^2\backslash SE(2)$ using finite Fourier coefficients. The convergence/error analysis of finite Fourier coefficients are investigated. Conditions are established for the finite Fourier coefficients to converge to the Fourier coefficients. The matrix forms of the finite transforms are discussed. The implementation of the discrete method to compute numerical approximation of $SE(2)$-convolutions with functions which are radial in translations are considered. The paper is concluded by discussing capability of the numerical scheme to develop fast algorithms for approximating multiple convolutions with functions which are radial in translations.