Quaternionic inverse Fourier Transforms on LCA Groups

On the (dis)continuity of the Fourier transform of measures

In this paper, we will study the continuity of the Fourier transform of measures with respect to the vague topology. We show that the Fourier transform is vaguely discontinuous on R, but becomes continuous when restricting to a class of Fourier transformable measures such that either the measures, or their Fourier transforms are equi-translation bounded. We discuss continuity of the Fourier transform in the product and norm topology. We show that vague convergence of positive definite measures implies the equi translation boundedness of the Fourier transforms, which explains the continuity of …

Titchmarsh theorem associated with QFT

In this paper, we prove both of the Titchmarsh theorems associated with Two-Sided Quaternionic Fourier Transform and we conclude about Short-Time Two-Sided Quaternionic Fourier Transform.

关于加权 Wiener-Lévy 定理：欧几里得空间上的类似形式、LCA 群上的强逆定理及其在调制空间中的应用

On the weighted Wiener-Lévy theorem: analogue on Euclidean space, strong converse on LCA group and applications to modulation spaces

本文在加权 Fourier 代数中证明了 Wiener-Lévy 定理的强逆定理及其在欧几里得空间和局部紧阿贝尔群上的类似形式，并应用于调制空间等以揭示色散 PDE 动力学中的非线性特征。

分数阶双侧四元数Dunkl变换与Heisenberg型不等式

The Fractional Two-Sided Quaternionic Dunkl Transform and Heisenberg-Type Inequalities

本文研究了二维分数阶双侧四元数Dunkl变换的定义与性质，建立了经典Heisenberg型不确定性原理的广义形式，并推广证明了适用于任意矩的高阶Heisenberg型不等式，最后推导出双侧分数阶四元数Fourier变换的相应不确定性原理。

Some results on the lattice parameters of quaternionic Gabor frames

Gabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the…

Gordon's Conjectures: Pontryagin-van Kampen duality and the Fourier transform in hyperfinite setting

Using the ideas of E.I. Gordon we present and farther advance an approach, based on nonstandard analysis, to simultaneous approximations of locally compact abelian groups and their duals by (hyper)finite abelian groups, as well as to approximations of various types of Fourier transforms on them by the discrete Fourier transform. Combining some methods of nonstandard analysis and additive combinatorics we prove the three Gordon's Conjectures which were open since 1991 and are crucial both in the formulations and proofs of the LCA groups and Fourier transform approximation theorems.

关于傅里叶变换的延拓

On extension to Fourier transforms

本文估计了LCA群中有限点集的最佳延拓算子范数，并据此证明无限闭子集上不存在到傅里叶变换空间的有界线性延拓算子，推广了Graham的紧集结果。

刻画第二可数紧致阿贝尔群上的Kirkwood-Dirac正性

Characterizing the Kirkwood-Dirac positivity on second countable LCA groups

本文研究了第二可数紧致阿贝尔群上Kirkwood-Dirac正性的性质，揭示了其与量子力学经典部分的关系，并给出了连通紧致阿贝尔群的几何描述。

Two wavelet multipliers and Landau-Pollak-Slepian operators on locally compact abelian groups associated to right-$H$-translation-invariant functions

By using a coset of closed subgroup, we define a Fourier like transform for locally compact abelian (LCA) topological groups. We studied two wavelet multipliers and Landau-Pollak-Slepian operators on locally compact abelian topological groups associated to the transform and show that the transforms are $L^p$bounded linear operators, and are in Schatten p-class for $1\leq p\leq \infty$. Finally, we determine their trace class and also obtain a connection with the generalized Landau-Pollak-Slepian operators.

非对易里斯变换--维数自由界和Fourier乘子

Noncommutative Riesz transforms -- Dimension free bounds and Fourier multipliers

在群von Neumann代数上得到了条件负长度函数的非交换里斯变换的无量纲估计.这包括泊松半群，超越巴克里的结果在交换设置。我们的证明受到了Pisier方法和交叉积的一个新的Khintchine不等式的启发。新的估计包括与$\mathbb{R}^n$中的分数拉普拉斯算子相关的里斯变换（其中迈耶猜想失败）或与自由群的字长相关。Lust-Piquard在LCA群上的离散拉普拉斯算子的工作也以几种方式推广。在傅立叶乘子的上下文中，我们将证明Hörmander-Mihlin乘子是我们的里斯变换的Littlewood-Paley平均。这在欧几里德和（最显著的）非对易环境中是非常令人惊讶的。作为应用，我们给出了新的Sobolev/Besov型光滑条件.我们给出的Sobolev型条件是对经典Sobolev型条件的改进，并得到了无量纲常数。我们的结果适用于任意幺模群。

双边 Clifford Dunkl 变换与 Miyachi 定理

The Two-Sided Clifford Dunkl Transform and Miyachi's Theorem

本文提出双边 Clifford Dunkl 变换，建立其反演与 Plancherel 公式，给出平移算子表达式，并证明 Miyachi 定理的类比，推广经典调和分析结果。

齐次Banach空间作为$M(G)$上的Banach卷积模

Homogeneous Banach spaces as Banach convolution modules over $M(G)$

本文通过构造精细有界均匀单位分割，避免结构理论和Haar测度，建立了有界测度空间$M(G)$在局部紧Abel群$G$的齐次Banach空间上的自然卷积模结构，并证明了这些空间是本质$L^1(G)$-模。

关于局部紧 Abel 群上的 Wiener 引理

On Wiener's Lemma on locally compact abelian groups

该研究借助 Fourier 分析与 Følner 序列理论，将 Wiener 引理推广到 LCA 群上的测度，并给出 R^d 与 T^d 上 Bochner–Riesz 平均版本。

非离散局部紧阿贝尔群上的广义群代数和广义测度代数

Generalized group algebras and generalized measure algebras on non-discrete locally compact abelian groups

本文在非离散局部紧阿贝尔群上推广了群代数和测度代数，定义了广义傅里叶变换，证明其像为自然BSE与BED的Banach函数代数，并研究了Segal代数、局部有界元素及乘子代数。