The ridgelet transform of distributions

分布的方向Stockwell变换

Directional Stockwell transform of distributions

本文提出方向Stockwell变换，结合方向短时傅里叶变换与ridgelet变换，证明了其Parseval恒等式和连续性，并建立了在Lizorkin分布空间上的分布框架。

紧支撑分布的Weyl变换

The Weyl transform of a compactly supported distribution

该研究刻画了$\mathbb{R}^{2n}$上紧支撑分布的Weyl变换的性质，建立了其$p$次幂可迹性与傅里叶变换$p$次幂可积性、紧算子性与傅里叶变换无穷远消失性的等价关系。

Tempered distributions and Fourier transform on the Heisenberg group

The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on $\H^d$is no longer a function on $\H^d$ : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Four…

Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition

In this paper, we study the class of tempered distributions whose Fourier transform is a translation bounded measure and show that each such distribution in $\mathbb{R}^d$ has order at most $2d$. We show the existence of the generalized Eberlein decomposition within this class of distributions, and its compatibility with all previous Eberlein decompositions. The generalized Eberlein decomposition for Fourier transformable measures and properties of its components are discussed. Lastly, we take a closer look at the absolutely continuous spectrum of measures supported on Meyer sets.

The short-time Fourier transform of distributions of exponential type and Tauberian theorems for shift-asymptotics

We study the short-time Fourier transform on the space $\mathcal{K}_{1}'(\mathbb{R}^n)$ of distributions of exponential type. We give characterizations of $\mathcal{K}_{1}'(\mathbb{R}^n)$ and some of its subspaces in terms of modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform.

分数阶辛傅里叶变换及其在函数与分布空间上的作用

Fractional symplectic Fourier transforms and their actions on function and distribution spaces

该研究探讨分数阶傅里叶变换的辛类比，通过Bargmann变换和Wigner分布给出其表示，证明其在多类函数与分布空间上的同胚性质，并证实其在Weyl演算相关的Schatten-von Neumann符号类上保持范数。

分布的方向短时分数阶傅里叶变换

The directional short-time fractional Fourier transform of distributions

本文提出了方向短时分数阶傅里叶变换（DSTFRFT），证明了其Parseval等式与重构公式，并基于算子连续性在缓增分布空间上建立了分布框架与去奇异化公式。

Images of function and distribution spaces under the Bargmann transform

We consider a broad family of test function spaces and their dual (distribution) space. The family includes Gelfand-Shilov spaces, a family of test function spaces introduced by S. Pilipovic. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions.

Fourier transform and rigidity of certain distributions

Let $E$ be a finite dimensional vector space over a local field, and $F$ be its dual. For a closed subset $X$ of $E$, and $Y$ of $F$, consider the space $D^{-ξ}(E;X,Y)$ of tempered distributions on $E$ whose support are contained in $X$ and support of whose Fourier transform are contained in $Y$. We show that $D^{-ξ}(E;X,Y)$ possesses a certain rigidity property, for $X$, $Y$ which are some finite unions of affine subspaces.

Moment Asymptotic Expansions of the Wavelet Transforms

Using distribution theory we present the moment asymptotic expansion of continuous wavelet transform in different distributional spaces for large and small values of dilation parameter $a$. We also obtain asymptotic expansions for certain wavelet transform.

二次相位傅里叶变换域中的高级Wigner分布与模糊函数：数学基础与实际应用

Advanced Wigner Distribution and Ambiguity Function in the Quadratic-phase Fourier Transform Domain: Mathematical Foundations and Practical Applications

针对非平稳信号处理需求，本文提出二次相位傅里叶变换域的高级Wigner分布与模糊函数（AWDQ/AAFQ），推导其关键数学性质，仿真验证其在线性调频信号检测中性能优于传统及现有同类方法。

Wigner Distribution and associated uncertainty principles in the framework of octonion Linear Canonical Transform

The most recent generalization of octonion Fourier transform (OFT) is the octonion linear canonical transform (OLCT) that has become popular in present era due to its applications in color image and signal processing. On the other hand the applications of Wigner distribution (WD) in signal and image analysis cannot be excluded. In this paper, we introduce novel integral transform coined as the Wigner distribution in the octonion linear canonical transform domain (WDOL). We first propose the definition of the one dimensional WDOL (1DWDOL), we extend its relationship with 1DOLCT and 1DOFT. Then …

Fourier transforms of spherical distributions on compact symmetric spaces

In our previous articles "A local Paley-Wiener theorem for compact symmetric spaces", Adv. Math. 218 (2008), 202--215, and "Fourier series on compact symmetric spaces" (submitted) we studied Fourier series on a compact symmetric space M=U/K. In particular, we proved a Paley-Wiener type theorem for the smooth functions on M, which have sufficiently small support and are K-invariant, respectively K-finite. In this article we extend those results to K-invariant distributions on M. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base…

The support theorem for the complex Radon transform of distributions

The complex Radon transform $\hat F$ of a rapidly decreasing distribution $F\in\mathscr{O}_C^{\prime}(\mathbb{C}^n)$ is considered. A compact set $K\subset\mathbb{C}^n$ is called linearly convex if the set $ \mathbb{C}^n \setminus K$ is a union of complex hyperplanes. Let $\hat K$ denote the set of complex hyperplanes which meet $K$. The main result of the paper establishes the conditions on a linearly convex compact $K$ under which the support theorem for the complex Radon transform is true: from the relation $\hbox{supp}(\hat F)\subset\hat K$ it follows that $F\in\mathscr{O}^{\prime}_C(\math…

The Signed Cumulative Distribution Transform for 1-D Signal Analysis and Classification

This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [ACHA 45 (2018), no. 3, 616-641] to arbitrary (signed) signals on $\overline{\mathbb{R}}$. We present both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity, linear separability and others. Finally, we describe a metric in transform space, and demo…

若干分布及其傅里叶变换的选择及其在磁共振成像中的应用

A Selection of Distributions and Their Fourier Transforms with Applications in Magnetic Resonance Imaging

本文以数学视角严谨介绍MRI中常见分布及其傅里叶变换，明确拓扑空间与定义，涵盖泊松求和公式及高斯函数变换，并力求自成体系。

Signed Cumulative Distribution Transform for Parameter Estimation of 1-D Signals

We describe a method for signal parameter estimation using the signed cumulative distribution transform (SCDT), a recently introduced signal representation tool based on optimal transport theory. The method builds upon signal estimation using the cumulative distribution transform (CDT) originally introduced for positive distributions. Specifically, we show that Wasserstein-type distance minimization can be performed simply using linear least squares techniques in SCDT space for arbitrary signal classes, thus providing a global minimizer for the estimation problem even when the underlying signa…

一种基于非分离核二次相位傅里叶变换的新型二维维格纳分布框架

A Novel Two-Dimensional Wigner Distribution Framework via the Quadratic Phase Fourier Transform with a Non-Separable Kernel

本文提出二维非分离二次相位维格纳分布（2D-NSQPWD），基于2D-NSQPFT框架推广经典维格纳分布，推导其关键性质并验证在多分量2D-LFM信号上交叉项抑制和定位的优越性能。