分布理論的建立
發(fā)布時間：2018-10-26 11:06
【摘要】：分布是廣義函數的泛函定義,它是在物理學和數學自身發(fā)展的背景下產生的。1936年,索伯列夫引入了廣義函數概念,他稱為有限階連續(xù)線性泛函。約十年之后,施瓦茲再次引入了廣義函數的泛函定義——分布,并建立了分布理論。這一理論不僅為近現(xiàn)代物理學的研究奠定了基礎,而且在數學各分支領域中有著廣泛應用,如偏微分方程、群表示論等。本文在原始文獻及其相關研究文獻的基礎上,利用文獻分析、歷史研究和比較研究的方法,以“為什么數學”為切入點,細致考察了施瓦茲提出分布概念、建立分布理論的過程、原因及其影響,取得了以下研究成果：1.探究出施瓦茲關于偏微分方程的廣義解工作激發(fā)他把古典函數概念推廣為卷積算子。然而,當他定義卷積算子的傅里葉變換時,施瓦茲碰到了無法克服的困難。因此,他開始另選新的數學對象來推廣經典函數概念。狄拉克函數實質上是一個測度,它能夠被看成質點的質量分布這一事實啟發(fā)施瓦茲在引入測度泛函定義的基礎上把經典函數概念推廣為測度,物理學中“多層”的定義則進一步激勵他把測度推廣為分布,從而他最終把古典函數概念推廣為分布。2.通過細致研究施瓦茲的分布工作發(fā)現(xiàn)：在布爾巴基學派結構數學觀念的影響下,施瓦茲考察了分布空間的結構;在泛函和對偶思想的幫助下,他定義了分布的各種運算,如導數、乘積和卷積等。從施瓦茲的工作中窺探出他的工作方式具有“一般化”和“抽象化”,這順應了20世紀數學發(fā)展的特征。3.揭示出施瓦茲想要求解卷積方程的目標,探究出他求解卷積方程的一般策略。被布爾巴基學派“代數化”之后,在卷積定理的啟示下,施瓦茲想要通過傅里葉變換把卷積方程轉化為代數方程,從而實現(xiàn)卷積方程的求解。正是這一思想指導著他考察了分布的卷積、傅里葉變換、乘法和除法,而定義分布的傅里葉變換則是他引入施瓦茲空間的原因所在。4.在全面考察索伯列夫及其廣義函數工作的基礎上分析出：雖然索伯列夫的廣義函數工作比施瓦茲早了近十年,但是他未能成為廣義函數理論奠基者是由其科研興趣、學術傳統(tǒng)、時代背景和歷史使命等因素共同所導致。5.剖析出以下原因使得施瓦茲能夠成功創(chuàng)建分布理論：首先是泛函分析的成熟、拉東測度的引進、韋伊的卷積工作以及施瓦茲早期關于局部凸拓撲向量空間的研究成果等數學工具的鋪墊；其次是他的布爾巴基學派背景,這不僅使他學到了結構數學的思想,而且他被“代數化”了；再者就是他求解卷積方程這一目標的激勵；還有就是索伯列夫為其留下了獨立的創(chuàng)作空間。6.指出在分布理論的基礎上,施瓦茲的大膽猜想、埃倫普里斯和馬爾格朗日的證明以及赫爾曼德爾的努力使常系數線性偏微分方程獲得了完整理論。
[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.
【學位授予單位】：西北大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O177
本文編號：2295520
[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.
【學位授予單位】：西北大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O177
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