用非标准离散函数和差商定义新广义函数(英文)

Defining New Generalized Functions by Nonstandard Discrete Functions and Difference Quotients

在非标准分析框架下 ,用离散函数定义新广义函数 ,用差商定义其导数 .对 Schwartz广义函数以及更广的 Gevrey超广义函数 ,文章证明了广义导数可以用差商表示 .此外还给出了此新广义函数和 Sobolev理论的关系 .

By using nonstandard analysis, we define new generalized functions as discrete functions, and their derivatives are defined as difference quotients. For Gevrey′s ultradistributions, including Schwartz′ distributions, we prove that difference quotients are indeed good replacements of generalized derivatives. Relations of our new generalized functions with Sobolev theory are presented. It is expected that this theory will be useful for nonlinear partial differential equations with distributional data, via difference method.