超复Fourier核的封闭表达式

发布时间：2018-06-16 02:00

本文选题：Clifford分析 + 超复Fourier变换　；参考：《哈尔滨工业大学》2017年博士论文

【摘要】：Fourier变换是目前称为Fourier分析的数学分支中的核心概念。经典Fourier变换不仅与其它数学分支,如偏微分方程、数论、表示论、数学物理有深刻的联系,而且在工程问题中有大量的应用,它已经成为科学研究中一个基本工具。Clifford分析为定义真正的高维Fourier变换,而不仅仅是一维Fourier变换的张量积提供了理论基础。近年来,Clifford-Fourier变换吸引了很多人兴趣。利用李代数sl2的表示理论与群对称性,研究者定义了推广的Clifford-Fourier变换,给出了满足特定条件的积分变换的完全分类。但是,人们仅获得了少数几种情形下积分核的明确表达式及其有界性。与sl2相相关的Fourier变换还有Dunkl变换、(k,a)-Fourier变换及其在Clifford代数框架下的推广等。这些变换积分核的明确表达式仍有待研究。特别的,近30年来,Dunkl核及其对称化Dunkl-Bessel函数的积分表示已经成为一个广泛关注的问题,其与Markov过程、随机矩阵、Calogero-Moser-Sutherland型量子多体系统、共形群的极小表示理论都有深刻的联系。本文主要针对以上几类Fourier核进行了研究,建立了基于Laplace变换的计算Clifford-Fourier变换积分核封闭表达式的新方法,得到了二面体群情形下的Dunkl核及其对称化、(k,a)-Fourier核的表达式,并对这些积分核精确的界进行了估计。本文主要研究内容包括以下几个方面:1.利用Clifford分析单演函数论,特殊函数和Laplace变换等工具,研究了 Clifford-Fourier核及其推广在Laplace域中的封闭表达式;得到了积分核的显明表达式与平面波分解;定义并计算了偶数维核的生成函数;对推广的Clifford-Fourier核的增长速度进行了估计;确定了可以生成有界Fourier核的多项式。2.考虑到双曲空间和欧氏空间不同的几何特性,在双曲空间中定义了推广的Helgason-Fourier变换;使用分数阶积分建立了 Clifford-Fourier积分核在Laplace域中的表达式与双曲空间中Fourier核之间的联系,导出了偶数维积分核的明确表达式及其形式生成函数。3.针对Dunkl核与(k,a)-Fourier核,利用Poisson核和Fourier核之间的关系,计算了推广Fourier核在Laplace域中的表达式;研究了(0,a)-Fourier核、二面体群情形下Dunkl核及其对称化的积分表达式,分析了当参数是整数时,这两类Fourier核封闭表达式的存在性;然后利用数学归纳法,特殊函数等技巧对(0,2/n)-Fourier变换积分核的界进行了估计。
[Abstract]:Fourier transform is a core concept in the branch of mathematics called Fourier analysis. Classical Fourier transform is not only closely related to other branches of mathematics, such as partial differential equation, number theory, representation theory, mathematical physics, but also has a large number of applications in engineering problems. It has become a basic tool in scientific research. Clifford analysis provides a theoretical basis for defining the real high-dimensional Fourier transform, not just the tensor product of one-dimensional Fourier transform. In recent years Clifford-Fourier transform has attracted a lot of interest. Using the representation theory of lie algebra sl2 and group symmetry, the authors define the generalized sl2 Fourier transform and give a complete classification of integral transformation satisfying certain conditions. However, the explicit expression of integral kernel and its boundedness are obtained only in a few cases. The Fourier transform related to sl2, as well as the Dunkl transform and its extension in the framework of sl2 algebras, are discussed in this paper. The explicit expressions of these transform integral kernels still need to be studied. In particular, the integral representation of Dunkl kernel and its symmetric Dunkl-Bessel function has become a widespread concern in the last 30 years. It is closely related to Markov processes, the stochastic matrix Calogero-Moser-Sutherland quantum multi-body system and the minimal representation theory of conformal groups. In this paper, several Fourier kernels mentioned above are studied, and a new method for calculating the closed expressions of Clifford-Fourier transform integral kernels based on Laplace transform is established. The expressions of Dunkl kernels and their symmetrically symmetric kernels are obtained in the case of dihedral groups. The exact bounds of these integral kernels are estimated. The main contents of this paper include the following aspects: 1. By using Clifford analysis function theory, special function and Laplace transform, the Clifford-Fourier kernel and its generalized closed expression in Laplace domain are studied, and the explicit expression of integral kernel and plane wave decomposition are obtained. The generating function of even-dimensional kernel is defined and calculated, the growth rate of the generalized Clifford-Fourier kernel is estimated, and the polynomial .2. which can generate bounded Fourier kernel is determined. Considering the different geometric properties of hyperbolic space and Euclidean space, the generalized Helgason-Fourier transform is defined in hyperbolic space, and the relation between Clifford-Fourier integral kernel in Laplace domain and Fourier kernel in hyperbolic space is established by using fractional integral. The explicit expression of even dimensional integral kernel and its form generating function. 3 are derived. In this paper, the expression of generalized Fourier kernel in Laplace domain is calculated by using the relation between Poisson kernel and Fourier kernel, and the integral expression of Dunkl kernel and its symmetrization in the case of dihedral group is studied, and when the parameter is an integer, the integral expression of Dunkl kernel and its symmetrization is studied. The existence of these two kinds of closed expressions of Fourier kernels is obtained and the bounds of the integral kernels of the two Fourier kernels are estimated by means of mathematical induction and special functions.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O174.2

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