分布理论的建立

发布时间：2018-10-26 11:06

【摘要】：分布是广义函数的泛函定义,它是在物理学和数学自身发展的背景下产生的。1936年,索伯列夫引入了广义函数概念,他称为有限阶连续线性泛函。约十年之后,施瓦兹再次引入了广义函数的泛函定义——分布,并建立了分布理论。这一理论不仅为近现代物理学的研究奠定了基础,而且在数学各分支领域中有着广泛应用,如偏微分方程、群表示论等。本文在原始文献及其相关研究文献的基础上,利用文献分析、历史研究和比较研究的方法,以“为什么数学”为切入点,细致考察了施瓦兹提出分布概念、建立分布理论的过程、原因及其影响,取得了以下研究成果：1.探究出施瓦兹关于偏微分方程的广义解工作激发他把古典函数概念推广为卷积算子。然而,当他定义卷积算子的傅里叶变换时,施瓦兹碰到了无法克服的困难。因此,他开始另选新的数学对象来推广经典函数概念。狄拉克函数实质上是一个测度,它能够被看成质点的质量分布这一事实启发施瓦兹在引入测度泛函定义的基础上把经典函数概念推广为测度,物理学中“多层”的定义则进一步激励他把测度推广为分布,从而他最终把古典函数概念推广为分布。2.通过细致研究施瓦兹的分布工作发现：在布尔巴基学派结构数学观念的影响下,施瓦兹考察了分布空间的结构;在泛函和对偶思想的帮助下,他定义了分布的各种运算,如导数、乘积和卷积等。从施瓦兹的工作中窥探出他的工作方式具有“一般化”和“抽象化”,这顺应了20世纪数学发展的特征。3.揭示出施瓦兹想要求解卷积方程的目标,探究出他求解卷积方程的一般策略。被布尔巴基学派“代数化”之后,在卷积定理的启示下,施瓦兹想要通过傅里叶变换把卷积方程转化为代数方程,从而实现卷积方程的求解。正是这一思想指导着他考察了分布的卷积、傅里叶变换、乘法和除法,而定义分布的傅里叶变换则是他引入施瓦兹空间的原因所在。4.在全面考察索伯列夫及其广义函数工作的基础上分析出：虽然索伯列夫的广义函数工作比施瓦兹早了近十年,但是他未能成为广义函数理论奠基者是由其科研兴趣、学术传统、时代背景和历史使命等因素共同所导致。5.剖析出以下原因使得施瓦兹能够成功创建分布理论：首先是泛函分析的成熟、拉东测度的引进、韦伊的卷积工作以及施瓦兹早期关于局部凸拓扑向量空间的研究成果等数学工具的铺垫；其次是他的布尔巴基学派背景,这不仅使他学到了结构数学的思想,而且他被“代数化”了；再者就是他求解卷积方程这一目标的激励；还有就是索伯列夫为其留下了独立的创作空间。6.指出在分布理论的基础上,施瓦兹的大胆猜想、埃伦普里斯和马尔格朗日的证明以及赫尔曼德尔的努力使常系数线性偏微分方程获得了完整理论。
[Abstract]:Distribution is a functional definition of a generalized function, which is produced under the background of the development of physics and mathematics itself. In 1936, Soberlev introduced the concept of generalized function, which he called finite order continuous linear functional. About ten years later, Schwartz introduced the functional definition of generalized function, distribution, and established the distribution theory. This theory not only lays a foundation for the study of modern physics, but also has been widely used in various branches of mathematics, such as partial differential equations, group representation theory and so on. On the basis of the original literature and its related research documents, using the methods of literature analysis, historical research and comparative study, this paper makes a careful study of the concept of distribution put forward by Schwartz, taking "why mathematics" as the starting point. The process, cause and influence of establishing distribution theory have obtained the following research results: 1. It is found that Schwartz's work on generalized solution of partial differential equations motivates him to generalize the concept of classical function to convolution operator. However, when he defined the Fourier transform of convolution operators, Schwartz encountered insurmountable difficulties. Therefore, he began to choose a new mathematical object to generalize the concept of classical function. Dirac function is essentially a measure. The fact that it can be regarded as the mass distribution of particles inspired Schwartz to generalize the concept of classical function as a measure on the basis of introducing the definition of measure functional. The definition of "multilayer" in physics further encourages him to generalize the measure to distribution, so he finally generalizes the concept of classical function to distribution. Through careful research on the distribution of Schwartz, it is found that under the influence of the idea of structural mathematics of the Bourbaki school, Schwartz examines the structure of the distribution space; With the help of functional and dual theory, he defines the operations of distribution, such as derivative, product and convolution. From the work of Schwartz, we can see that his working style is "general" and "abstract", which conforms to the characteristics of the development of mathematics in the 20th century. 3. The goal of Schwartz's solution to convolution equation is revealed, and his general strategy for solving convolution equation is explored. Inspired by the convolution theorem, Schwartz wants to transform the convolution equation into the algebraic equation by Fourier transform, so as to solve the convolution equation. It is this idea that guides him to examine the convolution, Fourier transform, multiplication and division of distribution, and the Fourier transform that defines distribution is the reason why he introduced Schwartz space. On the basis of a comprehensive review of Soberlev and his work on generalized functions, it is found that although Soberlev's generalized functions work nearly ten years earlier than Schwartz's, he failed to become the founder of the theory of generalized functions because of his interest in scientific research and academic tradition. Background of the times and historical mission and other factors together. 5. 5. The main reasons are as follows: firstly, the maturity of functional analysis and the introduction of Rato measure. Wye's convolution work and Schwaz's earlier research results on locally convex topological vector space, etc. Secondly, his background of Bourbachian school, which not only made him learn the thought of structural mathematics, but also was "algebraic", moreover, it was the motivation of his goal of solving convolution equation. And Soberlev left his own creative space. 6. It is pointed out that on the basis of distribution theory, Schwaz's bold conjecture, the proof of Ellen Plis and Margrad and the efforts of Helmand make the complete theory of linear partial differential equations with constant coefficients obtained.
【学位授予单位】：西北大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177

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