The Quaternion Domain Fourier Transform and its Properties

So far quaternion Fourier transforms have been mainly defined over \({\mathbb{R}^2}\) as signal domain space. But it seems natural to define a quaternion Fourier transform for quaternion valued signals over quaternion domains. This quaternion domain Fourier transform (QDFT) transforms quaternion valued signals (for example electromagnetic scalar-vector potentials, color data, space-time data, etc.) defined over a quaternion domain (space-time or other 4D domains) from a quaternion position space to a quaternion frequency space. The QDFT uses the full potential provided by hypercomplex algebra in higher dimensions and may moreover be useful for solving quaternion partial differential equations or functional equations, and in crystallographic texture analysis. We define the QDFT and analyze its main properties, including quaternion dilation, modulation and shift properties, Plancherel and Parseval identities, covariance under orthogonal transformations, transformations of coordinate polynomials and differential operator polynomials, transformations of derivative and Dirac derivative operators, as well as signal width related to band width uncertainty relationships.     

Banach空间值鞅变换的有界性及其应用

本文给出关于Banach空间值鞅变换算子有界性的一种新的处理方法,得到一系列带有广泛性的结果,并应用鞅变换算子的有界性刻画了Banach空间的一致光滑性和一致凸性,使得许多已有文献中的结论成为本文的特例.     

随机变量变换分布的若干推论及应用

给出了随机变量变换分布的三个推论,这些推论提供了在不同变换下求二维随机变量的函数的概率密度的计算公式,实例应用表明,这些公式应用简便,灵活,实用．     

Berezin Transform, Mellin Transform and Toeplitz Operators

We consider functions of the form f₁[`(g)]<sub>1</sub>+h{f_1\bar g_1+h} in the range of the Berezin transform B, where f <sub>1</sub> and g <sub>1</sub> are holomorphic on the unit disk \mathbb D{\mathbb D}, and h is either harmonic or of the form f<sub>2</sub>[`(g)]₂{f_2\bar g_2} for some holomorphic functions f ₂ and g ₂ on \mathbb D{\mathbb D}. First, by using the Mellin transform, we complement Ahern’s Theorem (Ahern in J Funct Anal 215:206–216, 2004) by proving that if u ? L¹{u\in L^1} and B(u) is harmonic, then u is harmonic. Secondly, we extend Ahern’s Theorem when h is harmonic, and give very precise relations between f ₁ and f ₂, g ₁ and g ₂ when h=f₂[`(g)]₂{h=f_2\bar g_2} and g ₂(z) = z ⁿ with n ≥ 1. Finally, some applications of our results to the theory of Toeplitz operators are discussed.     

On the Bargmann Transform and the Wigner Transform

We observe that the Bargmann transform arises from the polardecomposition of a simple restriction operator. By virtue ofthe Bargmann transform, the Wigner transform is imbedded inthe geometric symmetry group of phase space. From this imbedding,we easily obtain that the spectrum of the Wigner transform consistsof {e^ik/12:k=0, 1, ..., 23}. 1991 Mathematics Subject Classification81R30, 47B38, 81S30.     

傅立叶变换与小波变换的比较教学

本文针对小波变换教学中小流变换概念理解困难的问题,提出了一种比较教学方法,通过分析小波变换与傅立叶变换之间的联系,并从四个方面进行对比,清楚地描述了小波变换的本质,从而对加深对小波变换的理解。     

An Operator Transform from Class A to the Class of Hyponormal Operators and its Application

In this paper, we shall give an operator transform from class A to the class of hyponormal operators. Then we shall show that and in case T belongs to class A. Next, as an application of we will show that every class A operator has SVEP and property (β).     

推广约当引理以及Laplace变换与Fourier变换的对应关系

单变量复变函数积分中用到的约当引理是在lim||f（Reiφ）||＝0（0≤≤π）的条件下使用的，而实际上约当引理可以在比较宽松的条件下就可使用，被积函数f（z）除在z的上半平面（Ⅰmz≥0）有有限个孤立奇点外，处处解析，且对p＞0只要，则这里z＝Rei，CR为上半平面的开弧半圆围道，利用推广约当引理可以证明Laplace变换实际上是对应的其复变量函数的Fourier变换。     

Scale-Invariant Transform

Scale Invariant Transforms are defined for both one- and two-dimensioned input functions. These have the desirable properties of linearity and invariance to scale change of the input.     

On the Windowed Fourier Transform and Wavelet Transform of Almost Periodic Functions

We present the windowed Fourier transform and wavelet transform as tools for analyzing persistent signals, such as bounded power signals and almost periodic functions. We establish the analogous Parseval-type identities. We consider discretized versions of these transforms and construct generalized frame decompositions. Finally, we bring out some relations with shift-invariant operators and linear systems.     

Segal－Bargmann－Hall Transform and Geometric Quantization

Using geometric methods, Hall has proved that the Segal-Bargmann transform for a con-nected Lie group K of compact type is an isometric isomorphism [H1] and is unique when Kis simply connected [H7]. Furthermore, Hall considered geometric quantization of T~*(K), K'scotangent bundle [H9]. Using the vertical polarization and a natural Khler polarization obtainedby identifying T~*(K) with the complexified group KC, Hall concluded that the pairing map be-tween the two Hilbert Spaces induced by these two polarizations coincides with the generalizedSegal-Bargmann transform C_t (up to constant).     

傅里叶变换与处处连续无处可微函数

本文介绍如何在数学分析课程中引入傅里叶变换以及利用傅里叶变换说明魏尔斯特拉斯函数的无处可微性.     

Estimates of Oscillations for the Conjugate Hardy Transform and for the Calderon Transform

For the conjugate Hardy transform and for the Calderon transform, the BLO-norms are calculated and two-sided estimates of the norms in the BMO are obtained, most of which are exact. Bibliography: 9 titles.     

由一元连续小波变换得到的二元连续小波变换及重构公式

讨论了L2(R2)空间中连续小波变换,分别得到由一元变换函数构造二元变换函数的二元小波变换及重构公式,得到重构公式在L2(R2)中范数收敛意义下成立的条件.     

The Clifford-Fourier Transform

A pair of Clifford-Fourier transforms is defined in the framework of Clifford analysis, as operator exponentials with a Clifford algebra-valued kernel. It is a genuine Clifford analysis construct, which is shown to be a refinement of the classical multi-dimensional Fourier transform. An adequate operational calculus is developed.     

Uncertainty Principles for the Continuous Dunkl Gabor Transform and the Dunkl Continuous Wavelet Transform

In this paper we consider the Dunkl operators T _j , j = 1, . . . , d, on and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl translation operator, by proceeding as mentioned in [20] by C.Wojciech and G. Gigante. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl continuous wavelet transform introduced by K. Trimèche in [18], an analogous of this result for the Dunkl continuous wavelet transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous wavelet transform) is proved.      

The Kontorovich-Lebedev Transform

It is pointed out that difficulties can occur with the Kontorovich-Lebedevtransform when expressed in terms of Hankel functions and anew form is proposed which overcomes these difficulties.