关于SU(2)上的快速傅里叶变换
On the Fast Fourier Transform on SU(2)
AI总结：
本文针对非阿贝尔紧李群SU(2)上的谱分析计算挑战，提出一种基于Cooley-Tukey分治策略的快速傅里叶变换算法，通过欧拉角离散化、二维FFT和雅可比多项式递归性质，显著优于直接傅里叶变换。
摘要：
特殊酉群SU(2)在量子力学、理论物理和球面信号处理中的对称性描述中起着基础性作用。本文针对这一非阿贝尔紧李群上的谱分析计算挑战，提出了SU(2)上的傅里叶变换（FT），并开发了一种受经典Cooley-Tukey分治方案启发的快速傅里叶变换（FFT）算法。我们的方法使用欧拉角高效离散化群，对角度变量应用二维FFT，并利用雅可比多项式的递归性质。我们提供了计算复杂度分析，证明基于FFT的方法显著优于直接计算FT。该算法为理解SU(2)上FFT的实现奠定了基础，是弯曲流形和量子系统高性能计算应用中数值模拟和高级数据分析的关键组成部分。
英文摘要：
The special unitary group SU(2) plays a fundamental role in the description of symmetries in quantum mechanics, theoretical physics, and spherical signal processing. In this paper, we address the computational challenges of performing spectral analysis on this non-abelian compact Lie group. We present the Fourier Transform (FT) on SU(2) and develop a Fast Fourier Transform (FFT) algorithm inspired by the classical Cooley-Tukey divide-and-conquer scheme. Our approach efficiently discretizes the group using Euler angles, applying a two-dimensional FFT on the angular variables and exploiting the recursive properties of Jacobi polynomials. We provide an analysis of the computational complexity, demonstrating that our FFT-based method significantly outperforms the direct computation of the FT. This algorithm serves as a foundational tool for understanding the implementation of the FFT on SU(2), a key component in numerical simulations and advanced data analysis for high-performance computing applications on curved manifolds and quantum systems.
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