分布的乘积与一类具有分布系数的常微分方程
A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients
摘要：
我们构造了L. Hörmander in [L. Hörmander，{ \it线性偏微分算子的分析I}（Springer-Verlag，1983）]。新的乘积定义在分段光滑函数$f: \bkR \to \bkC$及其所有（分布）导数的向量空间${\mathcal A}(\bkR)$中。它是结合的，满足莱布尼茨规则，并再现了${\mathcal A}(\bkR)$中正则分布函数的通常逐点乘积。赋予这个乘积，空间${\mathcal A}(\bkR)$就成为广义函数的微分结合代数。通过在新的${\mathcal A}(\bkR)$ -设置工作，我们确定了一种方法，用于将具有通解$ψ$的常线性微分方程转化为具有通解$χ_Ω ψ$的常线性微分方程，其中$χ_Ω$是某个指定区间$Ω\subset \bkR$的特征函数。
英文摘要：
We construct a generalization of the multiplicative product of distributions presented by L. Hörmander in [L. Hörmander, {\it The analysis of linear partial differential operators I} (Springer-Verlag, 1983)]. The new product is defined in the vector space ${\mathcal A}(\bkR)$ of piecewise smooth functions $f: \bkR \to \bkC$ and all their (distributional) derivatives. It is associative, satisfies the Leibniz rule and reproduces the usual pointwise product of functions for regular distributions in ${\mathcal A}(\bkR)$. Endowed with this product, the space ${\mathcal A}(\bkR)$ becomes a differential associative algebra of generalized functions. By working in the new ${\mathcal A}(\bkR)$-setting we determine a method for transforming an ordinary linear differential equation with general solution $ψ$ into another, ordinary linear differential equation, with general solution $χ_Ω ψ$, where $χ_Ω$ is the characteristic function of some prescribed interval $Ω\subset \bkR$.
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