Extracting a hidden procedure by non-commutative harmonic analysis

作为建立利用非交换调和分析提取隐藏过程的方法的一步，本研究提出了几种从定义在非交换群上的信号的傅立叶展开中提取其重要特征的方法，例如：(1)通过有效的卷积在二面体群上对给定信号进行频域运算的方法(2)将作为三维旋转群中离散网格的信号的域变换为更精细网格的插值法，利用来自信号的平滑分量对频域中的某些分量的贡献的集中(3)一种方法，其提供表示n个对象的排列组上的函数在多大程度上受1、2、…、n的对象标签的改变的影响的定量测量。

As a step toward establishment of the way to extract a hidden procedure by using non-commutative harmonic analysis, this research presented several methods which extract some important features of signals defined over a non-commutative group from their Fourier expansion, such as: (1) A method which operates a given signal over a dihedral group in its frequency domain, by an efficient convolution (2) An interpolation method which transform the domain of a signal which is a discrete mesh in the3-dimensional rotation group into a more finer mesh, exploiting the concentration of the contribution from smooth component of the signal to certain components in the frequency domain (3) A method that provides a quantitative measure representing to what extent a function over the group of permutations of n objects is affected by the change of the labeling of objects by 1, 2, ..., n.