Full algebra of generalized functions and non-standard asymptotic analysis
Full algebra of generalized functions and non-standard asymptotic analysis
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广义函数的全代数和非标准渐近分析
DOI：
10.1007/s11813-008-0008-y
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
H. Vernaeve
中科院分区：
文献类型：
--
作者：
T. Todorov;H. Vernaeve
构造了具有Schwartz分布空间的典范嵌入的广义函数代数，给出了Schwartz分布乘法问题的一个解决方案，证明了该代数的标量集是代数闭域，而不同于哥伦布理论中的零因子环.我们证明了一个在科伦博理论中不成立的Hahn-Banach扩张原理。我们在我们的理论和非标准分析之间建立了联系，从而间接地回答了科伦博提出的一个问题。本文在广义函数的科伦博理论和非标准分析之间架起了一座桥梁。
We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.