具有动态检验函数和与不连续函数相乘的分布
Distributions with dynamic test functions and multiplication by discontinuous functions
摘要：
根据施瓦茨不可能性定理，两个分布相乘一般是不可能的。然而，通常需要将分布乘以不连续函数，而不是任意分布。在本文中，我们构建了一个空间的分布，其中的一般操作的乘法由一个不连续的函数定义，连续的，交换的，结合的莱布尼茨产品规则。在新的分布空间中，经典的δ函数$δ_τ$扩展为一族δ函数$δ_τ^α$，依赖于\textit{shape} $α$。我们证明了经典分布空间$\mathcal D'$中的赫维赛德函数和δ函数的乘积的各种已知定义成为新分布空间中乘法的特殊情况，并提供了新分布空间对最优控制理论中出现的常微分方程的应用.此外，我们比较了我们的方法的Schwartz分布理论与Colombeau的广义函数代数，其中的两个分布的乘法的一般操作的定义的方法。
英文摘要：
As follows from the Schwartz Impossibility Theorem, multiplication of two distributions is in general impossible. Nevertheless, often one needs to multiply a distribution by a discontinuous function, not by an arbitrary distribution. In the present paper we construct a space of distributions where the general operation of multiplication by a discontinuous function is defined, continuous, commutative, associative and for which the Leibniz product rule holds. In the new space of distributions, the classical delta-function $δ_τ$ extends to a family of delta-functions $δ_τ^α$, dependent on the \textit{shape} $α$. We show that the various known definitions of the product of the Heaviside function and the delta-function in the classical space of distributions $\mathcal D'$ become particular cases of the multiplication in the new space of distributions, and provide the applications of the new space of distributions to the ordinary differential equations which arise in optimal control theory. Also, we compare our approach of the Schwartz distribution theory with the approach of the Colombeau generalized functions algebra, where the general operation of multiplication of two distributions is defined.
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