An existence and uniqueness result about algebras of Schwartz distributions

Nuno Costa Dias, Cristina Jorge, João Nuno Prata
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引用次数: 1

Abstract

Abstract We prove that there exists essentially one minimal differential algebra of distributions $$\mathcal A$$ A , satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l’impossibilité de la multiplication des distributions, 1954], and such that $$\mathcal C_p^{\infty } \subseteq \mathcal A\subseteq \mathcal D' $$ C p A D (where $$\mathcal C_p^{\infty }$$ C p is the set of piecewise smooth functions and $$\mathcal D'$$ D is the set of Schwartz distributions over $$\mathbb R$$ R ). This algebra is endowed with a multiplicative product of distributions, which is a generalization of the product defined in [N.C.Dias, J.N.Prata, A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients, 2009]. If the algebra is not minimal, but satisfies the previous conditions, is closed under anti-differentiation and the dual product by smooth functions, and the distributional product is continuous at zero then it is necessarily an extension of $$\mathcal A$$ A .
Schwartz分布代数的一个存在唯一性结果
摘要证明了存在一个极小的分布微分代数 $$\mathcal A$$ A,满足Schwartz不可能结果中的所有性质[L]。Schwartz, Sur l ' impossible it de la multiplication des distribution(1954),等等 $$\mathcal C_p^{\infty } \subseteq \mathcal A\subseteq \mathcal D' $$ C p∞(其中 $$\mathcal C_p^{\infty }$$ C p∞是分段光滑函数和的集合 $$\mathcal D'$$ D '是Schwartz分布的集合 $$\mathbb R$$ R)。这个代数被赋予了分布的乘法积,它是在[N.C.]中定义的乘积的推广Dias, J.N.Prata,一类具有分布系数的常微分方程的乘积[j]。如果代数不是极小的,但满足上述条件,在反微分和光滑函数的对偶积下是封闭的,并且分布积在零处连续,那么它必然是一个扩展 $$\mathcal A$$ 选A。
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