Analytic functionals on the complex sphere

复球面上的解析泛函

基本信息

批准号：
11640181
负责人：
MORIMOTO Mitsuo
金额：
$ 1.73万
依托单位：
International Christian University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640181/
关键词：
Complex sphere Analytic functional Harmonic function Entire function of exponential type Fourier-Borel transformation Reproducing kernel Lie ball Dual Lie ball フーリェ・ボレル変換

项目摘要

In connection with that analytic functions and analytic functionals on the compex sphere are as boundary values of holomorphic harmonic functions in the Lie ball, we treat in [1] an integral rep-resention of Cauchy-Hua type. The entire functions which are the Fourier-Borel image of analytic functionals on the complex sphere are eigenfunctions of the Laplacian operator. In [2] we represent these eigenfunctions as an integral over the complex light cone. In [3] we treat the Fourier transfor-mation on the Hardy space of harmonic functions on the Lie ballIn [5] we study the reproducing kernel related with the complex sphere. In the papers [6] and [7] we discuss the relation between generalized functions, e.g., distributions, hyperfunctions, on the (real) sphere and solutions of the heat conduction equation. In [6] we study the one dimensional case, while in [7] we consider the n-dimensional sphere. These studies rely on the head investigator's results on the sphererical harmonic expansion of generalized functions on the real sphere. In [8] we study the double series expansion of holomorphic functions on the Lie ball, the dual Lie ball, or the complex Euclidean ball. [12] is a survey paper on the double series expansion. The last paper [13] shows there are a series of norms on the complex Euclidean space including the Lie norm, the dual Lie norm and the Euclidean norm. The result is given with calculationIn [11] we study how to define the Boehrnians on the sphere. The Boehmian is a generalized function usually defined on the Euclidean space by means of convolution. The convolution on the Euclidean space is a commutative operation but the convolution on the rotation group is not. We show two methods to overcome this difficulty

鉴于复球上的解析函数和解析泛函是李球中全纯调和函数的边界值，我们在[1]中将其视为Cauchy-Hua型的积分表示。复球面上解析泛函的 Fourier-Borel 图像的整个函数是拉普拉斯算子的本征函数。在[2]中，我们将这些本征函数表示为复杂光锥上的积分。在[3]中我们研究了李球上调和函数Hardy空间的傅立叶变换。在[5]中我们研究了与复球相关的再现核。在论文[6]和[7]中，我们讨论了（实）球体上的广义函数（例如分布、超函数）与热传导方程的解之间的关系。在[6]中我们研究一维情况，而在[7]中我们考虑n维球体。这些研究依赖于首席研究员关于真实球体上广义函数的球调和展开的结果。在[8]中，我们研究了李球、对偶李球或复欧几里得球上全纯函数的双级数展开。 [12]是一篇关于双级数展开的调查论文。最后一篇论文[13]表明，复欧几里得空间上存在一系列范数，包括李范数、对偶李范数和欧几里得范数。计算给出了结果。在[11]中我们研究了如何定义球面上的Boehrnians。波姆函数是通常通过卷积在欧几里德空间上定义的广义函数。欧几里得空间上的卷积是可交换运算，但旋转群上的卷积不是。我们展示了两种克服这个困难的方法