广义函数代数的整体理论
A global theory of algebras of generalized functions
摘要：
给出了一种几何方法来定义光滑流形$M$上的广义函数代数$\hat{\mathcal G}(M)$（Colombeau代数），该流形包含$M$上的分布空间${\mathcal D}'(M)$.基于方便向量空间中的微分学，我们得到了$\hat{\mathcal G}(M)$的一个内在构造。$\hat{\mathcal G}(M)$是一个{ \em微分}代数，它的元素具有关于任意光滑向量场的李导数.此外，我们构造了${\mathcal D}'(M)$到$\hat{\mathcal G}(M)$的一个正则线性嵌入，使得${\mathcal C}^\infty (M)$成为$\hat{\mathcal G}(M)$的忠实子代数.最后，它表明，这种嵌入交换与李衍生物。因此，$\hat{\mathcal G}(M)$在全局上下文中保留了局部理论的所有区别性质。
英文摘要：
We present a geometric approach to defining an algebra $\hat{\mathcal G}(M)$ (the Colombeau algebra) of generalized functions on a smooth manifold $M$ containing the space ${\mathcal D}'(M)$ of distributions on $M$. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of $\hat{\mathcal G}(M)$. $\hat{\mathcal G}(M)$ is a{\em differential} algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of ${\mathcal D}'(M)$ into $\hat{\mathcal G}(M)$ that renders ${\mathcal C}^\infty (M)$ a faithful subalgebra of $\hat{\mathcal G}(M)$. Finally, it is shown that this embedding commutes with Lie derivatives. Thus $\hat{\mathcal G}(M)$ retains all the distinguishing properties of the local theory in a global context.
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