Titchmarsh theorem associated with QFT

短三角和的$Ω$ -定理

$Ω$-theorem for short trigonometric sum

本文给出了经典E.C. Titchmarsh'离散方法（1934）在理论的黎曼$\zf$ -功能。也就是说，我们将证明第一个局部化的$Ω$ -定理短三角和。本文是参考文献\cite{4}的英文版。

Plancherel theorem and quaternion Fourier transform for square integrable functions

The quaternion Fourier transform (QFT), a generalization of the classical 2D Fourier transform, plays an increasingly active role in particular signal and colour image processing. There tends to be an inordinate degree of interest placed on the properties of QFT. The classical convolution theorem and multiplication formula are only suitable for 2D Fourier transform of complex-valued signal, and do not hold for QFT of quaternion-valued signal. The purpose of this paper is to overcome these problems and establish the Plancherel and inversion theorems of QFT in the square integrable signals space…

四元数Fourier定理与线性正则反演定理

Quaternion Fourier and Linear Canonical Inversion Theorems

四元数傅里叶变换（QFT）是研究彩色图像处理的重要工具之一。事实上，对QFT的深刻理解已经创造了作为整体而不是作为颜色分离的组件进行转换的彩色图像。此外，理解QFT为理解其他积分变换铺平了道路，例如四元数分数傅里叶变换（QFRFT），四元数线性正则变换（QLCT）和四元数Wigner-Ville分布。本文的目的是双重的：第一，提供一些关于四元数约束变差函数的理论背景。然后，我们应用它来推导四元数傅立叶和线性正则反演公式。其次，对四元数傅立叶和线性正则反演定理在绝对可积函数空间上的应用提供了一些指导。

Benedicks-Amrein-Berthier type theorem related to the two-sided Quaternion Fourier transform

The main objective of the present paper is to establish a new uncertainty principle (UP) for the two-sided quaternion Fourier transform (QFT). This result is an extension of a result of Benedicks, Amrein and Berthier, which states that a nonzero function in $L^1\left({\mathbb{R}}^2, {\mathbb{H}}\right)$ and its two-sided QFT cannot both have support of finite measure.

Miyachi's Theorem for the Quaternion Fourier Transform

The quaternion Fourier transform (QFT) satisfies some uncertainty principles similar to the Euclidean Fourier transform. In this paper, we establish Miyachi's theorem for this transform.

四元数的正交二维平面分裂与可控四元数Fourier变换

The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations

在\cite{Ell:1993}中引入双边四元数傅里叶变换（QFT），用于分析二维线性定常偏微分系统。在进一步的理论研究中，\cite{10.1007/s00006-007-0037-8, EH:DirUP_QFT}引入了一种特殊的四元数分裂，称为$\pm$分裂。在当前的\change{chapter}中，我们进一步分析了这种分裂，将其几何解释为\change{an} \emph{orthogonal 2D planes split}（OPS），并将其推广到$\H$到两个正交的2D分析平面的可自由操纵分裂。OPS分裂的新的一般形式使我们能够找到QFT对信号作用的新的几何解释。这项工作的第二个主要结果是各种\emph{new steerable forms}的QFT，他们的几何解释，并为每一种形式\change{,} OPS分裂定理，这使得快速和有效的数值实现与标准FFT软件。

The Continuous quaternion Algebra-Valued Wavelet Transform and the Associated Uncertainty Principle

The purpose of this article is to extend the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform (QFT). We study some fundamental properties of this extension such as scaling, translation, rotation, Parseval's identity, inversion theorem, and a reproducing kernel, then we derive the associated Heisenberg-Pauli-Weyl uncertainty principle UP. Finally, using the quaternion Fourier representation of the CQWT we generalize the logarithmic UP and Hardy's UP to the CQWT domain.