ABSTRACT

In this paper, after introducing the concepts of quaternionic dual group and the quaternionic valued character on locally compact abelian group $G^2$, the inverse of the quaternionic Fourier transform (QFT) on locally compact abelian groups is investigated. Due to the non-commutativity of multiplication of quaternions, there are different types of QFTs right, left and two-sided quaternionic Fourier transform. We focus on the right-sided quaternionic Fourier transform (RQFT) and two-sided quaternionic Fourier transform (SQFT). We establish the quaternionic Plancherel and inversion theorems for the square integrable quaternionic-valued signals on $G^2$, the space $L^2\left(G^2,\mathbb{H}\right)$, where G is a locally compact abelian group. Also RQFT on the space $L^2\left(G^2,\mathbb{H}\right)$ is studied. Furthermore relations between RQFT and SQFT are discussed. These results provide new proofs for the classical inverse Fourier transform, Plancherel theorem, etc. in $L^2\left(G\right)$.

摘要

本文在介绍了四元对偶群的概念和局部紧阿贝尔群的四元值特征后 $G^2$，研究了局部紧阿贝尔群上四元数傅里叶变换 (QFT) 的逆。由于四元数乘法的非对易性，QFT 有右、左和两侧四元数傅里叶变换的不同类型。我们专注于右侧四元数傅里叶变换 (RQFT) 和两侧四元数傅里叶变换 (SQFT)。我们建立了平方可积四元数值信号的四元普朗切雷尔定理和反演定理$G^2$， 空间 ${升}^2\left(G^2,\mathbb{H}\right)$，其中G是局部紧阿贝尔群。还有空间上的RQFT${升}^2\left(G^2,\mathbb{H}\right)$被研究。此外还讨论了 RQFT 和 SQFT 之间的关系。这些结果为经典傅里叶逆变换、Plancherel 定理等提供了新的证明。${升}^2\left(G\right)$.